I am mainly thinking about the group $\Gamma(N)$. A weakly modular form of weight one is a holomorphic function $f: \mathfrak{H} \to \mathbb{C}$ satisfying $$ f(\gamma \tau) = (c\tau+d)f(\tau), \qquad \gamma \in \Gamma(N), $$ and which is also "meromorphic at the cusps". It corresponds to an algebraic section of the Hodge line bundle $\underline{\omega}$ over $Y(N) = \mathfrak{H}/\Gamma(N)$.

I believe that, at least for small values of $N$, the Hodge line bundle is actually trivial (in other words, the universal elliptic curve over $Y(N)$ admits a Weierstrass equation). In particular, the trivialization of the $\underline{\omega}$ should correspond to some explicit weakly modular form of weight 1 with no zeros.

Where can I find explicit formulas for such modular forms? For instance, can they be defined by a series on $\mathfrak{H}$ (much like Eisenstein or Poincaré series)?

nothave order 1, 2 or 3. Over this space the universal elliptic curve has a Weierstrass equation. So we can pull back this universal Weierstrass equation to any modular curve. $\endgroup$