You can save your signature as a 'Generator' and set the editable fields which could be edited by your team members. For example: Name, Phone Number, Designation, etc.
You can invite your team members using the 'Invite' or 'Email All' option. Alternatively, you can also share the 'Signature Install URL' with your team members.
Your team members can edit the editable fields (without signing in/signing up) and install their signatures while you would still have the master access to all the signatures. You can review the signatures, edit or remove them.
Would your team members be required to install the signature again if the main template is changed?
Yes and No. If you use G Suite, the changes would be automatically made to the signatures without re-installing it. However, with all the other email clients, your team members would be required to install the email signature again.