How do I link outside emails to specific keywords?

for example if someone typed in a request for "dogs" it would automatically send the form submission to a specific email address for the person who handles the requests for "dogs".

You may setup conditions to send the notifications based on the user's response. Please refer to the guide below that can help you in achieving your requirement.

https://www.jotform.com/help/167-How-to-Send-Email-Based-on-User-s-Answer

Hope this information helps!