If you are able to run a compatible web browser on a Chromebook then you should be able to index using one.
Some compatible web browsers are Google Chrome, Mozilla Firefox and Microsoft Edge (Chrome based).
Its not so much whether its chrome or not.
Its more - how much RAM does it does it have - and what is your Internet bandwidth.
those two things will be the greatest factors in what your experience is with your chrome book as it relates to indexing and FamilySearch. (as with most any device) - as long as it supports major browsers.
I have not tried a Chromebook. But I found this information on FamilySearch.
Public Instructions: You can participate in web indexing from a desktop computer, laptop, and tablet device. You should be able to use Safari or Firefox as your web browser, though we recommend Google Chrome.
We support tablets with a minimum of a 1024x600 resolution. You will, of course, need an internet connection, as well as one of the following operating systems:
Other browsers and systems might work, but we might not be able to help you identify or troubleshoot any problems you might encounter.
Note: At this time, we are unable to support indexing from smart phones due to their small screen size.
Chromebook can be sluggish with overloads I'm told. . And some apps are not used easily.
But it is worth a try. If it is what you have, and you want to Index, what have you got to lose. Well, maybe the Chromebook.
Let me know. Enjoy.
@Lorrie Roemer Healy
As an aside, from the great information that 'Anitra' has already proffered ...
I am 'Tagging" this general 'Question' of yours in the "Group' of "Indexing Chat" in this Forum.
Perhaps, other members of that group have experience; and, can advise you, regarding with "Indexing" with the "Chromebook" Browser.
You should be able to. Using Chrome or another browser, go to: https://www.familysearch.org/indexing/my-indexing.
I use a chromebook for both indexing and family history research. I don't have any problem. (except occasionally when they do upgrades)