Hi there. The answer is very simple. You can just ask them if they cover travel costs. Really this is not something you should be afraid of.

Usually people are afraid to ask employers/potential employers questions like these due to fear of seeming greedy. A question like this will not put your job offer at risk.

Like with all things, the manner in which you ask will make all the difference.

Try being honest, and explain that you traveling to Washington will be a major financial burden and ask if they would be willing to cover the full or portion of the cost.