The typical and simplest approach for this is to use a portal since you have multiple related records.
But you can use a set of global fields in place of the fields shown here. Your Done button then performs a script that creates the new record in Requests and copies over the Primary key from your layout's current record as well as the data from each of these global fields.
It's also possible to set up an additional relationship to an added table occurrence of requests. This relationship uses a different match field in clients and has "allow creation.." enabled for this new occurrence of requests in the relationship. Use a script performed OnObjectEnter set on the popover's panel to clear this match field each time that the popover opens. Entering data into one of the fields creates the new related record and you would set up cancel to use Revert record to cancel the creation of a new requests record.