Yes this will be possible.
I would suggest integrating your WordPress websites search with Google search and then you can specify the site/s (in your cause Site B) you wish to search when a users performs a search on your current site (Site A)
Here is a awesome and simple tutorial a-z tutorial on how to set it all up: http://www.wpbeginner.com/wp-tutorials/how-to-add-google-search-in-a-wordpress-site/
Goodluck 🙂
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