Hey Simon,
I doubt if we can do it with the bubble’s searchbox element. Instead you can build a custom searchbox by using an Input for entering the search query and repeating group placed in a group focus for showing matched results. You can filter the results by email and on the Repeating group you can show the Name of the results.
You’ll have to build your own. I’d use a focus group that’s toggled when the inputs length is greater than 0
Stick the RG to the input
In the RG, an RG that uses the input as a query constraint against your users. Then you can display whatever part of the matched record that you’d like