Yes. Just like an internet service provider, a VPN can log all activity occurring on it. This includes which sites each user visits, and any actions they perform there.
Typically, corporate VPNs will log user activity on the VPN. Most personal VPN services promise not to.
Whether the VPN does log your internet traffic – or if it does but deletes it after a set period like 24 hours – is a matter of trust between you and the VPN service.