Can I upgrade my OS from High Sierra to Monterey (the latest?) on my early 2015 Macbook air? Version 10.13.6

Yes. See here: link-> How to upgrade to macOS Monterey - Apple Support

gdiddy789 wrote:

Thank you ... that was the perfect page!

But ...

When I used Time Machine to back up my info before upgrading, I didn't have enough memory room on my external hard drive. It said I needed 108 GB storage available.

Do you think I will need to purchased a larger storage device?

Probably.

Unless you can delete older backups from the hard drive and make enough room, you will need a larger drive.