Do I use the shift-control-c and shift-control-v method each time I need to copy a username and password from a vault’s keepass program?
Yes. Highlight entry in keepassxc, ctrl-c, ctrl-shift-c, switch to AppVM, clt-shift-v, ctrl-v. (Maybe change the keybinding for ctrl-shift-c, ctrl-shift-v to something requiring fewer keystrokes.)
Or I have just thought, do I need to install the Keepass program in a VM’s template for use in the VM and only have the database itself in the vault?
Install keepassxc in the template that the vault VM is based on. Only need to run keepassxc in the vault.