Yes and no. If you think of your current document as a template, you can save it with a different name to create a new document that's based on the current one. Whenever you want to create a document like that, you'll open your document in Word Online, go to File > Save As, and create a document that's a copy of the one you started with.
On the other hand, if you're thinking of the kind of templates you see by going to File > New, then no: you can't create those in Word Online.
Instead, do this:
If you have the Word desktop application, use the Open in Word command in Word Online to open the document in Word on your desktop.
From there, create a template. When you go to File > New in the Word desktop application, you'll be able to use your template for new documents. And if you store the documents online, you can edit them in Word Online.
Tip: We're continually updating Word Online based on input like yours. Tell us how we could improve this feature for you. At the bottom of the Word Online window click Help Improve Office.
Vote for an idea, or share a new one in our suggestion box at word.uservoice.com.