Replacement rear wheel to match an existing front wheel I found a discarded Mongoose Alta frame which has an front wheel marked Araya - Japan'.

The wheel reads 26 x 1.50 HEGP710, what rear wheel should I get?
Any regular mountain bike wheel with diameter 26 inches, rim width between 20-30 mm and a freewheel would likely to work. Cheap wheels in this category cannot be too wide for you, and absolutely most of them have 135 mm hub width and QR skewer to attach the wheel, which should be what your frame's rear dropouts will accept.

You will also need to buy matching tires (anything from 26" × 1.5" to 26" × 2.1" to your liking), inner tubes (almost any marked as 26" will do) and a cassette that fasts to the wheel's freehub. The choice of cassette depends on how many gears your rear derailleur has; for an 8-speed derailleur you will need a cassette with 8 cogs etc.