Ordering tickets online when already in a London apartment and have no printer?

an internet cafe? there are still some about, particularly in minority communities where they also offer internet telephoning.

a public library? plenty of them about with printers.

On occasion I've asked hotels for assistance with this type of thing. Several have been kind enough to let me e-mail them the .pdf of my ticket and they will print it for me at the front desk. You might enquire if this is an option with your hotel.

Edit: just realized you're in an apartment vs. hotel. That's a bit more tricky. Will you be carrying a smartphone or other device where you could present the tickets electronically? You'd need to e-mail or call the Hampton Court Palace offices of course to see if they will accept your ticket in that format.

If you are only going to Hampton Court (which is on the edge of London), just buy tickets on the day at the station (ticket machine or manned counter) or use your Oyster card if you have one (cheaper).
This is a suburban train and print-at-home tickets are not available.

In general, the reason to buy online tickets is to get cheaper advance purchase tickets. There is no point in buying online for travel on the same day, it is just an added level of complication.

I have been to Hampton Court Palace a couple of times in the height of the summer season and have never had to wait in much of a line at all. Maybe having pre-printed, skip-the-line tickets isn't terribly necessary for this destination? Others may have had a different experience...all I know is the two times I have been there and lines weren't an issue.

The last time we visited Hampton Court we waited in line 15 minutes at most.