Due to character restrictions the title terminology might not be quite right, by vector I effectively mean a given point on a given line that the circles are to be tangent to.
It feels like this should have some elegant solution, but I was unable to work though it with (free) wolfram alpha. I believe the answer should be able to be in the from of the radius in terms of x, y, and angle. Might also be able to be in terms of distance between givens, angle1, angle2.
The dotted arrow coming out of A is the first given vector and the arrow coming out of B pointing at E is the second. The large dotted circles are the circle pair we want the radius of. The rest is part of my attempts to figure out what I had to work with for solving the problem
Let the given tangents meet at $T$ with an angle $\theta$ (see figure). The lines perpendicular to the tangents at $A$ and $B$ meet at $Q$ with an angle $\pi-\theta$. Set: $a=AQ$ and $b=BQ$. The cosine rule applied to triangle $QCD$ gives then: $$ (a-r)^2+(b-r)^2+2(a-r)(b-r)\cos\theta=4r^2, $$ a quadratic equation which can be solved for $r$.
The case where the tangents are parallel is easier and is left to the reader.