Is it possible to do an iOS clean install without iTunes (I've sold my old MacBook)

For a “clean install”, you will need access to either a PC or Mac with the latest version of iTunes installed.

It is not possible to independently initiate a full reinstallation of iOS from your iDevice.

That shouldn’t happen. You should be able to restore the device using any computer running the latest version of iTunes.

Restore your iPhone, iPad, or iPod to factory settings - Apple Support

Before you connect your device to the friend’s computer, launch iTunes and go to iTunes>Preferences>Devices (Mac) or Edit>Preferences>Devices (Windows) and Check the box at the bottom of the window that says Prevent iPods, iPhones and iPads from automatically syncing. Click OK. Close iTunes. iTunes will not launch automatically when you connect your device.

Now you can connect your device and launch iTunes without syncing his content. After you Trust the computer, you can restore to factory settings using the instructions in the support article.

Do you have a friend/relative/co-worker with a computer?

LotusPilot is correct, so if you no longer have access to a computer, you can make an appointment at an Apple Store and ask them to restore the device for you.

Apple Store - Find a Store - Apple