I looked in Bob Dawson's
Romany in the Family which is a good starting point when looking at potential Traveller surnames and he lists Avery but not Gribble.
The reference is
1847 GLS which I think means he's seen a written instance of Avery in Gloucestershire in 1847 where the context identifies the person as being a member of the Traveller community.
Bristol was in Gloucestershire and Somerset so your Bristol reference is looking good.
If you can find a copy of
My Ancestors Were Gypsies by Sharon Sillers Floate in a library I'd suggest having a read of that to give you some ideas as to what you are looking for in relation to this Avery family.
I googled "avery 1847 gloucestershire" and it found a number of things so try that and have a look and see if any of those give you any useful information.
One of them was
https://www.gloucestershire.gov.uk/archives/learning-for-all/online-exhibitions/i-didnt-know-you-had-that/ which appears to have one Avery reference but I haven't read it.
You need to do a family tree for Hester (may also be written as Est(h)er) Avery and follow her siblings too to see if you can find any more evidence of Traveller ancestry.
Hope this helps.