When someone accelerates towards a light source, how does the light that's radiating off from it look? I suppose in an inertial frame, all the light should be moving off straight from the source, but I feel like somehow the light should curve due to the acceleration.
The light that the source emits will still travel in a straight line (light always travels in a straight line in an inertial frame). But as the source accelerates, the direction in which it emits light will change - see the answers to your other question.
A consequence is that, if you have a source that emits light in all directions when its at rest, then it will emit most of its light along its direction of motion when it moves at high velocity with respect to an stationary inertial observer. The result looks like this:
(source: wiki) This phenomenon is called relativistic beaming.
Also, the light will be Doppler-shifted (blueshifted along the direction of motion, redshifted along the opposite direction). The combined effect looks like this animation (although the animation shows what a moving observer sees from stationary sources, the same animation applies to what a moving source emits relative to stationary observers).