I have Workers that update project custom forms all the time. You need to grant contribute access and make sure that "edit custom form" is enabled. I personally do this via the template and grant contribute access to the entire team so no matter who works on the project they can do jump in and do the job. You can also do this via the access settings (on the project or the template) that when someone is assigned a task they also get contribute access to the project.
@BridgetJo2 I'm not quite following your question. On project template you can set the access level for individuals and teams instead of having to do it on every individual project. If you want someone to be able to edit a custom form they need a Worker license type as well as Contribute rights to the project or task. Within the Contribute rights, custom form needs to be toggled on. Use my screenshot to reference.