I was wondering if there was a way to switch out the VLAN ID from
VLAN1 to VLAN10 without impacting the connectivity of the end users?
First, you must remove the addressing from VLAN 1 before you could put that same addressing on VLAN 10 (interfaces cannot have the same or overlapping addressing).
Next, if that address is your gateway, then you will be removing the gateway (even for a few seconds), and that will impact any devices sending traffic to a different network. The hosts will probably have the MAC address of the VLAN 1 interface cached in their ARP tables as the gateway MAC address, and that will cause an impact.
Also, you must then change any switch interfaces using VLAN 1 to VLAN 10, and that will mean that you have an impact.
No matter what you do, it would probably be best to shut down all the VLAN 1 interfaces before starting, then remove the VLAN 1 addressing, add the addressing to VLAN 10, change the interfaces from VLAN 1 to VLAN 10, then enable the VLAN 10 interfaces. That will make sure that the interface ARP tables on the devices are cleared, and the devices will get the correct MAC addresses for the gateway.