Yes, there is and is rather called $t$-expansion rather than $q$-expansion - may be, though, you might want to replace your $\Omega$ by $$ \widehat{\bar{K_\infty}}\setminus K_\infty $$ where I denote by $K_\infty$ your completion $\mathbb{F}_q\big(\big(\frac{1}{T}\big)\big)$. This is the content of the paper On the coefficients of Drinfle'd modular forms by E-U. Gekeler, Inventiones Math. 93 (1988).

Yes, there is and is called $t$-expansion rather than $q$-expansion - may be, though, you might want to replace your $\Omega$ by $$ \widehat{\bar{K_\infty}}\setminus K_\infty $$ where I denote by $K_\infty$ your completion $\mathbb{F}_q\big(\big(\frac{1}{T}\big)\big)$. This is the content of the paper On the coefficients of Drinfel'd modular forms by E-U. Gekeler, Inventiones Math. 93 (1988).