Suggest you search old threads for Northern VA boarding. There must be dozens of them.
Please be aware the Washington DC area is the 7th largest metropolitian statistical area in the United States. We are populous. McLean is a high rise edge city that is no longer on the edge. Boarding there was possible in the 1960's but it has been decades since those days.
Ashburn is a bedroom suburb in Loudoun Country. You will be able to find boarding west of Ashburn in Leesburg and the hinterlands.
We're a horsey area, but land is at a premium here. It's a great location for eventing, except be prepared to pay $$$ wherever you go and spend time/money driving to the barn. Good luck!
I have quite a few friends who have done internships on the hill and with NGOs in DC. From what they've said, it doesn't seem like you'd have much time for a horse. If I were you I'd lease horse out for the summer and focus on the experience of the internship rather than stressing about getting to the barn every day.
"Choose to chance the rapids, and dare to dance the tides" - Garth Brooks
"With your permission, dear, I'll take my fences one at a time" - Maggie Smith, Downton Abbey
Where will you be interning and where will you be living? There are a few barns between McLean and Ashburn but they will be expensive. The Great Falls Horse Network may be a good place to post. My neighbor in Great Falls might take a summer boarder. PM me if you want their email or phone number.