Bike Nashbar AL-1 ,Raligh M50 , Schwinn Traveler , and others
8 speeds freewheels are hard to find , plus they are well known for breaking axles , ( to many gears ,not enough support ) . you will also need to redish the wheel . It doesn't matter to the derailleur how many gears it has ,what does is the shiftier and if it indexing or not . i hope this help.
'''96 Litespeed Catalyst, '05 Litespeed Firenze, '06 Litespeed Tuscany, '12 Surly Pacer, All are 3x8,9 or 10. It is hilly around here!
Simple answer, no. As bikeman715 mentioned 8-speed freewheels are unusual for good reason as they are wider than 6/7-speed freewheels, require a longer axle and tend to break them due to excess unsupported length.
You can either replace the 7-speed freewheel with a new one or get a new freehub rear wheel that will accept 8,9 or 10-speed cassettes. Your current rear derailleur or nearly any replacement will work fine if you use friction shifting. If you have index shifting you will need matching shifters for the number of cogs you choose.
I did some research to try to find out what I could do myself. From what I found out, hubs below Shimano Deore or Tiagra (like Altus, 2200) apparently are not worth it. I ended up buying a Shimano Tiagra rear hub (spacing is 130mm, Deore should be 135mm) on sale at niagaracycle and DT Swiss Champion 14g spokes (if I remember correctly, 287 and 289mm) with 12mm brass nipples. I also bought a WTB Freedom Ryder 23 rim on sale at eBikestop. Then you can always add rim tape or strip, spoke protector and wheel reflectors. There are videos on Youtube that show you how to build a wheel then you can have the tension adjusted at a shop. Someone here recommended putting beeswax on the spoke threads.