Non interactive threshold signature without bilinear pairing (is it possible)?
The scheme as written is insecure if an attacker can get the same message signed twice.

The honest parties will compute their $m_i$ deterministically, but the attacker doesn't have to.

The attacker will get $(m_i - c \cdot y_i) = p_i$ and $(m_i - c' \cdot y_i)= p_i'$ for each participant.

These equations can be solved to get $y_i = \frac{p_i - p_i'}{c' - c}$. Since the attacker knows all the values on the right-hand side, the attacker can get $t+1$ shares and recover $y$.

One possible solution to avoid this is to ensure that the same message is never signed twice.