I would try to wind up at the Foggy Bottom Metro station at dusk and take a taxi back to Union Stati

I will be arriving tomorrow at Union Station at 3 pm and my connecting train leaves at 10 pm. I have never been to DC. What would you recommend I use my short time to see? And how to get there. Medical issues airline mileage credit card prevent me from walking any long distances. Thanks in advance.
The Old Town Trolley tour of DC stops at Union Station, but since it stops at 6 I'm not sure it's worth your while. Taxis are expensive and difficult to catch at Union Station. The postal museum is across the street from Union Station. At 7:30, the Trolley does a nighttime tour, but it lasts 2-1/2 hours, airline mileage credit card so it wouldn't get you back until 10pm, and if it's a few minutes late you'd have problems.
I think you have fairly limited options. Your best bet might be to take the Metro to one of the Smithsonian museums and spend a couple of hours there until the museum closes, then have a bite to eat, and wait until after dark to go back to Union Station by taxi, doing a tour that way (you might be able to get a nice drive for under $40). However, I'd wait until after rush hour to take the taxi so you don't spend all your time stuck in traffic. You should be able to store your bags for a fee if you have a valid Amtrak ticket for same-day travel.
What constitutes a long distance? Could you visit a museum? The National airline mileage credit card Mall is very near Union Station, and you could take a taxi to the museum of your choice. Go to the Smithsonian website to see what's available. They're all free. You could return airline mileage credit card to Union Station for dinner.
Taxis are NOT difficult to get at Union Station IMO, especially during daylight hours. But you should also know that there is a Metro stop at Union Station as someone above mentioned and the entrance airline mileage credit card to it is via escalator.
I would definitely head to the National Mall, either by metro at Union Station or by taxi. That will give you maximum museum bang for your short visit. And a view of the US Capitol and the Washington Monument. If you're more interested in art than history and technology, make sure you look at a metro map to get the station closest airline mileage credit card to the National Gallery of Art.
I've never had trouble getting a cab at Union Station during regular business hours. And DC cabs are metered. I also disagree that the OP should necessarily wait until dark or after rush hour to try and get back to Union Station. Even in the heart of rush hour he/she would have plenty of time to get there from the National Mall. It's a very short distance.
Even if you aren't an art aficionado, the Smithsonian's National Portrait Gallery American Art Museum are routinely open until 7pm so you could visit after the Mall museums close at 5 or 5:30. They are spitting distance from the Gallery Place metro and the neighborhood, Penn Quarter, has many interesting restaurant options up and down the price scale. That neighborhood won't take you farther away from getting airline mileage credit card back to Union Station. Easy to catch a cab in general from the street, but if in doubt go across the street from the Portrait Gallery and catch one in front of the Hotel Monaco.
One quick note: the museums airline mileage credit card along the Mall are very large, and therefore the distances airline mileage credit card between them can be long for someone with walking issues. That's why I suggest planning in advance which one or two museums you'd really like to see, and perhaps cabbing from Union Station just to save steps, although the Metro is very easy and convenient, too.
I suggested waiting until after dark to return to Union Station because it gives the OP a chance to drive by the monuments at night, and if the OP takes a tour by taxi rather than just a ride to the station, then traffic is an issue. I'll stand by that recommendation.
I would try to wind up at the Foggy Bottom Metro station at dusk and take a taxi back to Union Station. Tell the driver you want to drive past the Lincoln Memorial and down the Mall rather than the shortest route. It probably won't be the first time he has done it.
I suggested waiting until after dark to return to Union Station because it gives the OP a chance airline mileage credit card to drive by the monuments at night, and if the OP takes a tour by taxi rather than just a ride to the station, then traffic is an issue. I'll stand by that recommendation.
obx, I assume that the OP would not want to do the "monument by taxi" tour as a round trip from Union Station (I am cheap) and s/he has a fair amount of time to kill. I also assume that the OP shares my horror of backtracking. Therefore, what is the closest Metro stop to the Lincoln Memorial? Foggy Bottom? Is it easy to get a taxi there? Yes. Can you take a taxi from there to Union Station along the Mall and see all the lighted monuments? Yes, and the Kennedy Center and the White House, too.