Searching sermon body (not just metadata)

Is there a way to search the sermon body for a text string?

Yes - one of the options of the Other Search in 10.0 is the ability to search your documents

There are details on using this search type at https://support.logos.com/hc/en-us/articles/8720049365389-Other-Search 

Note that in the current beta of 10.1, the Other search is going away and there is a specific search option for Docs - so, assuming this is part of the stable 10.1 release, the way to do this search will change but the functionality will effectively be the same.

Does "Other Search" exist on the web app?

No - this is not available on that platform

"Other" was an earlier desktop implementation that included a Document search , now available as Search > Docs.
But still not available on the web app!