I bit late but i i think it is proven there is no solutions, except for the special case 0° and side lengths 1, 1 and 0. Let us consider the triangle with a²+b²=c² and a = c sin(pi q) where q is the angle as a fraction of half a circle. So you are looking for a solution where a, b, c are integer and q is rational. So we first need to find a rational value for q where sin(pi q) is rational. According to https://math.stackexchange.com/questions/87756/when-is-sinx-rational#87768 this happens only for the well known case of 30°, so q=1/6 and a/c=1/2. However, in this case b=c/2 × sqrt(3) which is irrational, so with this angle we can never create integer side length.

Do you mean rational in radians, or in degrees?

It looks like there aren't any, by Niven's Theorem https://www.proofwiki.org/wiki/Niven%27s_Theorem