Is it possible to let someone upload files without creating an account?
Yes - it's possible to let someone upload files using an external upload request link.
This gives someone only the ability to add files, which will show up under the document request in the matter. They won't give access to any of the existing information in the matter.
These are the steps to create such a link:
Create a document request
You can create a matter and immediately add a document request to it (no need to go into the questionnaire and fill anything in):

Navigate to the external link screen:

Create an External Upload Link
Fill in the form to create the link:

Once created, click the "show link" button to reveal the specific link (these are valid for 7 days, and can be extended):

You can copy the link and email it to the person you want to request the file(s) from.
Using the external link
When they open the link they'll get to a screen like this:

Reviewing uploaded files
The "internal reference" will show up next to the upload date in the "documents" tab on the matter:

Frequently Asked Questions
Can the person with the link see anything else in the matter?
No. The external upload link only lets them add files, which show up under the document request on the matter. It gives no access to any existing information in the matter.
How long is the external upload link valid?
Links are valid for 7 days and can be extended.
How do I know who uploaded the files?
The internal reference you set on the link shows up next to the upload date on the Documents tab of the matter.