You can convert the Glock 22 to a .357 Sig by just buying a barrel. You can get a Glock OEM barrel which would be the G31 or an after market barrel from Lone Wolf Distributing, KKM, Storm Lake, Wilson, Barsto and EFK Fire Dragon. Most barrels are drop in and require no fitting.
I bought a Lone Wolf Distributing .357 Sig barrel for $110.00. It dropped right in with no fitting. It fits perfect and shoots great with 100% reliability. I recommend Lone Wolf Distributing barrels.
Here is my Glock 22 .357 Sig conversion.
I use the OEM Glock 22 magazines with the .357 Sig. It is a blast to shoot.