A group of Microsoft Products and technologies used for sharing and managing content, knowledge, and applications.
HI Alison,
Yes, you can. Planner actually uses a SharePoint Document Library to house the documents. a Document Library has a feature called Check-out, that you can enforce so that only one person can edit it at a time.
Here's the link to learn more:
Check out, check in, or discard changes to files in a library
You can directly access the Document Library by clicking on an attachment, then on the top-left, click on the Shared Documents to access the Document Library.
Then, go to the Library settings via the Cog button on the top-right of the library > Versioning settings > and select Require documents to be checked out before they can be edited? to Yes.
Regards,
Sheen