To see if you can upgrade to a 10 speed cassette in the rear, you need to check the rear dropout spacing first.

7-speeds were originally 126mm between the rear dropouts. Some manufacturers spaced the rear dropouts at 130mm (compatible with 8, 9, 10 speed cassettes), even on bikes that they supplied with 7-speed cassettes. My 97 Trek 1400 has a 7-speed cassette, and it has dropouts spaced at 130mm.

If you have a bike with a steel frame, although many do not recommend it, if your frame has 126mm rear dropout spacing, you could try having a shop reset the rear dropouts to 130mm, but there is some risk that the joints will pop, and the frame will be ruined (the shop will not be responsible if the frame is ruined). If you have an aluminum or carbon fiber bike, resetting the rear dropout spacing is not possible.

After determining if the rear droputs are wide enough, you need to check your rear wheel axle length to see if you can change the 7-speed freehub body to a 8,9, 10 freehub body and redish the rear wheel, or if the rear axle is 130, and has 8, 9, 10 freehub body and spacer for use with the 7-speed cassette.

I would think that if you need to buy a freehub body, and pay a shop to redish the rewheel, it may be less expensive to buy a replacement rear wheel.

If you have a newer rear wheel, and has a 10speed freehub, you're all set.

Remember that you will need to buy a new rear derailleur, new chain, and a new right side (rear) shifter (I don't know if your 7-speed front derailleur and left side shifter will work, or if they need to be replaced).

There are still plenty of 7-speed replacement and upgrade parts available. While your LBS may not actively be restocking 7-speed parts as they are sold, you may be able to swing a sweet deal with a bike shop that has the parts sitting in inventory gathering dust. You can also find 7-speed parts all day on eBay.