For finite $p$-groups one has the Lazard correspondance. This gives an equivalence of categories between $p$-groups of class less than $p$ ("Lazard groups") and Lie rings of class less than $p$ whose additive groups are $p$-groups. Given such a $p$-group $G$, its Lazard corespondent is a Lie ring defined on the same underlying set as $G$, with addition and bracket product defined in terms of the group operations as follows:

$$ x +_L y = x y [x,y]^{-1/2} [y,[x,y]]^{7/12}[x,[y,x]]^{5/12} [y,[y,[y,x]]]^{5/8} [y,[x,[y,x]]]^{1/2}[x,[x,[y,x]]] ^{3/8}\ldots $$

$$[x,y]_L = [x,y][y,[x,y]]^{-1/2}[x,[y,x]]^{1/2}[y,[y,[y,x]]]^{-1/3}[y,[x,[y,x]]]^{-1/4}[x,[x,[y,x]]]^{-1/3}\ldots $$

(these are related to the Baker-Campbell-Hausdorff formula). Example: for $G$ the extra-special $p$-group of order $p^3$ and exponent $p>2$, the corresponding Lie ring is the 3-dimensional Heisenberg Lie algebra over $\mathbb{F}_p$. See for example Khukhro's book on $p$-automorphisms of $p$-groups or Lazard's Sur les groups nilpotents et les anneaux de Lie.

There is a correspondence of representations here, but only in small dimension. Any characteristic $p$ rep can be thought of as a homomorphism into the group of upper unitriangular $n\times n$ matrices. For $n\leq p$ this has class $\leq p-1$, and its associated Lie ring is isomorphic, via the log map, to the Lie algebra of strictly upper triangular matrices. So functoriality gives a correspondence of representations of dimension $\leq p$.

It is worth pointing out that in characteristic p, it is not usually possible to find a Lie algebra with the same representation theory as a given group. This is because all Lie algebras have their cohomology finitely generated over the subring generated by degree two elements,while few groups have this property. However there are miraculous special cases such as the dihedral group of order eight whose modular group algebra is actually isomorphic to a universal enveloping algebra.