VLANs are lower than IP so you don’t need a router to have a VLAN, but you will need a router to get packets between the networks. I don’t think a WiFi repeater works. You likely need separate WiFi client and AP devices so you can put your WiFi on a different channel. Otherwise you’re probably halving your WiFi performance when connecting to the other network over the same airwaves.

Unless you can convince the other network to route your IP addresses, this setup will give you another layer of NAT and may cause problems with online games.