I think there's a couple questions here. 1. How can we estimate $q$ for any point inside the support and 2. how can we estimate $q$ for any point outside the support.
For 1, I'd try taking some inspiration from the Radon transformation, in particular something like : https://en.wikipedia.org/wiki/Algebraic_reconstruction_technique

Discretize the support into a whole bunch of points $x_i$; we'll try to recover $q(x_i)$. Use the values of the integrals to get linear constraints for $\sum_j a_{ij} q(x_j) = \int_{V_i} q$. This will be under-determined, so add in a smoothness penalty to bound $\frac{q(x_i)-q(x_j)}{x_i-x_j}$ for neighboring $x_i$, $x_j$, and whatever other bounds you thing might be fun. Plug it all into your favorite solver and see what comes out.

For 2., that's going to be a lot harder, but maybe once you have a solution for 1. you can try kriging.

I think there's a couple of questions here:

1. How can we estimate $q$ for any point inside the support and

2. how can we estimate $q$ for any point outside the support.
   For 1, I'd try taking some inspiration from the Radon transformation, in particular, something like Algebraic reconstruction

Discretize the support into a whole bunch of points $x_i$; we'll try to recover $q(x_i)$. Use the values of the integrals to get linear constraints for $\sum_j a_{ij} q(x_j) = \int_{V_i} q$. This will be under-determined, so add in a smoothness penalty to bound $\frac{q(x_i)-q(x_j)}{x_i-x_j}$ for neighboring $x_i$, $x_j$, and whatever other bounds you thing might be fun. Plug it all into your favorite solver and see what comes out.

For 2., that's going to be a lot harder, but maybe once you have a solution for 1. you can try kriging.