Need assistance identifying new wheel options for Araya VP-20 26x1.5 ETRTO 559

'VP-20' is likely just a manufacturer model number or similar.

On mountain bikes wheel and tire diameters are often referred to in inch sizes: 26", 27.5" and 29". Those measurements are nominally the diameter of the inflated tire. What they really mean is the tires fit wheel rim bead-seat diameters of 559mm, 584mm and 622mm. The ERTRO specification gives the rim bead-seat diameter in millimeters, in your case 559mm

1.5 is some form if measurement of the width of the rim, but it is unclear whether it is the outside total width or the internal bead seat diameter width. In any case a tire of a certain width will fit on a range of rim widths. See answers to this question What is the maximum or minimum tire width I can fit on my bicycle. You need a wheel with a rim bead-seat width that will work with the tire width you want to use.

Remember that you also need to consider specifications of the hubs. A front wheel with have a dropout spacing of 100mm. Rear wheels have different width depending on how old the bike is but could be 126mm, 130mm or 135mm.

An older bike may have a freewheel (sprockets and ratchet mechanism in a single unit that threads onto the hub). You can replace this with a wheel with a 'freehub' (ratchet mechanism in the hub, separate sprockets in a 'cassette') if the dropout spacing is 130 or 135mm.