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[EDIT] My original incomplete answer is below, but Sasha Premet has provided the complete reference, the table of 4.6 in Jantzen's 1991 paper. Jantzen recalls that he computed various cases in types $E_\ell$ not covered by the computer calculations of his Oregon colleagues Gilkey and Seitz, using his amazing Sum Formula. This formula applies for all primes but is usually quite daunting to compute. Probably his non-computer results are completely correct, but skeptics should feel free to duplicate them by whatever means. It's definitely challenging to get any conceptual insight for these small primes. (Revisions to my lecture notes are posted here.)

This question probably hasn't been fully answered for the exceptional types $E_6, E_7, E_8$. In general, the linkage principle (developed fully in Jantzen's book Representations of Algebraic Groups) ensures that for $p$ "large enough", each fundamental weight $\varpi$ lies in the closure of the lowest $p$-alcove for the affine Weyl group (of Langlands dual type) and thus the Weyl module $V(\varpi)$ is indeed simple. Here "large enough" depends on the Coxeter number and the coefficients of the highest short root, so the problem reduces to a computational one for a definite range of primes. The bad primes (possibly 2, 3, 5 for exceptional types) cause special problems, but apart from those the fundamental modules for types $G_2, F_4$ are well-behaved.

In Chapter 4 of my 2006 survey Modular Representations of Finite Groups of Lie Type (LMS Lecture Notes Series 326), I tried to cover all results known to me, with complete references. For example, Table 1 on page 37 summarizes results computed by Gilkey-Seitz for $F_4$ which confirm that fundamental Weyl modules are simple when $p>3$. Symplectic groups have been thoroughly examined by Premet-Suprunenko, Foulle, McNinch. And so on.

The fundamental modules for types $E_\ell$ probably haven't all been studied well enough to settle all cases. However, the few fundamental weights which are minuscule pose no problem since the weights of the corresponding Weyl modules form a single orbit under the Weyl group and thus the modules remain simple for all $p$. But otherwise the linkage principle just gives a bound on how much computation is needed. Probably nothing unusual happens for good primes, but that's not clearly documented yet.

One other theoretical remark: Lusztig's conjecture should handle all primes greater than the Coxeter number in a uniform way, but it has not yet been proved for this optimal lower bound. Meanwhile direct computation using existing algorithms (which go back to early work by Wong and Burgoyne) seems to be the only alternative.

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[EDIT] My original incomplete answer is below, but Sasha Premet has provided the complete reference, the table of 4.6 in Jantzen's 1991 paper. Jantzen recalls that he computed various cases in types $E_\ell$ not covered by the computer calculations of his Oregon colleagues Gilkey and Seitz, using his amazing Sum Formula. This formula applies for all primes but is usually quite daunting to compute. Probably his non-computer results are completely correct, but skeptics should feel free to duplicate them by whatever means. It's definitely challenging to get any conceptual insight for these small primes.

This question probably hasn't been fully answered for the exceptional types $E_6, E_7, E_8$. In general, the linkage principle (developed fully in Jantzen's book Representations of Algebraic Groups) ensures that for $p$ "large enough", each fundamental weight $\varpi$ lies in the closure of the lowest $p$-alcove for the affine Weyl group (of Langlands dual type) and thus the Weyl module $V(\varpi)$ is indeed simple. Here "large enough" depends on the Coxeter number and the coefficients of the highest short root, so the problem reduces to a computational one for a definite range of primes. The bad primes (possibly 2, 3, 5 for exceptional types) cause special problems, but apart from those the fundamental modules for types $G_2, F_4$ are well-behaved.

In Chapter 4 of my 2006 survey Modular Representations of Finite Groups of Lie Type (LMS Lecture Notes Series 326), I tried to cover all results known to me, with complete references. For example, Table 1 on page 37 summarizes results computed by Gilkey-Seitz for $F_4$ which confirm that fundamental Weyl modules are simple when $p>3$. Symplectic groups have been thoroughly examined by Premet-Suprunenko, Foulle, McNinch. And so on.

The fundamental modules for types $E_\ell$ probably haven't all been studied well enough to settle all cases. However, the few fundamental weights which are minuscule pose no problem since the weights of the corresponding Weyl modules form a single orbit under the Weyl group and thus the modules remain simple for all $p$. But otherwise the linkage principle just gives a bound on how much computation is needed. Probably nothing unusual happens for good primes, but that's not clearly documented yet.

One other theoretical remark: Lusztig's conjecture should handle all primes greater than the Coxeter number in a uniform way, but it has not yet been proved for this optimal lower bound. Meanwhile direct computation using existing algorithms (which go back to early work by Wong and Burgoyne) seems to be the only alternative.

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This question probably hasn't been fully answered for the exceptional types $E_6, E_7, E_8$. In general, the linkage principle (developed fully in Jantzen's book Representations of Algebraic Groups) ensures that for $p$ "large enough", each fundamental weight $\varpi$ lies in the closure of the lowest $p$-alcove for the affine Weyl group (of Langlands dual type) and thus the Weyl module $V(\varpi)$ is indeed simple. Here "large enough" depends on the Coxeter number and the coefficients of the highest short root, so the problem reduces to a computational one for a definite range of primes. The bad primes (possibly 2, 3, 5 for exceptional types) cause special problems, but apart from those the fundamental modules for types $G_2, F_4$ are well-behaved.

In Chapter 4 of my 2006 survey Modular Representations of Finite Groups of Lie Type (LMS Lecture Notes Series 326), I tried to cover all results known to me, with complete references. For example, Table 1 on page 37 summarizes results computed by Gilkey-Seitz for $F_4$ which confirm that fundamental Weyl modules are simple when $p>3$. Symplectic groups have been thoroughly examined by Premet-Suprunenko, Foulle, McNinch. And so on.

The fundamental modules for types $E_\ell$ probably haven't all been studied well enough to settle all cases. However, the few fundamental weights which are minuscule pose no problem since the weights of the corresponding Weyl modules form a single orbit under the Weyl group and thus the modules remain simple for all $p$. But otherwise the linkage principle just gives a bound on how much computation is needed. Probably nothing unusual happens for good primes, but that's not clearly documented yet.

One other theoretical remark: Lusztig's conjecture should handle all primes greater than the Coxeter number in a uniform way, but it has not yet been proved for this optimal lower bound. Meanwhile direct computation using existing algorithms (which go back to early work by Wong and Burgoyne) seems to be the only alternative.