I have a campaign I created and shared my content on. One of my players has created a character. When they go to edit it later, it has a circle with a line through it and "Join with this character" is grayed out. They only have the free account with no subscription.

Is this a bug, or do they need a subscription to edit and play the character they created?

To clarify, they cannot edit or view the character sheet.

That sounds like they're trying to rejoin the campaign with the character, which is totally unrelated to editing the character. Try this:

From the megamenu at the top of the page, they should hover over collections, then click on 'My characters.' That should bring them to a list of their characters that they've built in DDB. The character should be listed there, and from there they should be able to view or edit. It will also indicate whether or not that character is already in a campaign. Characters can only be in one campaign. From this screen they should be able to click 'leave campaign' to remove it from any other campaign it might be in. Then they can try re-joining the campaign they should be in. But either way, that screen should always allow them to access any characters they've created on their account.