For general integral:

$$\int_0^{\infty } \frac{J_0(q \rho ) \exp (L-q z) (R J_1(q R)-a J_1(q a))}{q} \, dq$$

I think that probably does not have finite closed-form expression in terms of very large class of special functions. It can't be solved analytically.

but numerically can be computed:

    R=0.05;
    L=0.1;
    a=0.0005; 
    \[Rho] = 1;
    z = 1;
    NIntegrate[ BesselJ[0, q \[Rho]] Exp[
       L - q z] (R BesselJ[1, q R] - a BesselJ[1, q a])/q, {q, 
      0, \[Infinity]}]

  (*0.000976668*)