Yep, you can use a VPN with Clash Royale. Plenty of players do, mostly to keep their connection safer on public Wi-Fi or locked-down networks.
Supercell’s Terms of Service don’t straight-up say “no VPNs,” but they do say: don’t cheat, exploit the game, or use sketchy tools. So using a VPN for privacy is fine, but using it to game the system is where you’ll run into trouble.