You can apply to jobs there, and sign up for internship notification - though I've been on that list for two years now and haven't gotten anything
The Cincinnati tournament and the Washington Kastles have job listings on teamworkonline.com more often, which a lot of MLB teams use as well if you're interested in that. WTT and the US Open Series/US Open are more likely to hire for the summer for internships. The USTA US Open office is HQ'ed in Westchester I believe, so you'll be working out of the city for most of the summer until the tournament rolls around
The WTA is headquartered in St. Petersburg, Florida, and the ATP in Ponte Vedra Beach, so I assume you'd have to live/work around there. I'm personally looking for a legal/law-related internship, and the firm the WTA uses is a large one in DC, which is probably out of my reach at the moment anyway.