While you can not control this 100%… there are a few options..
Try using the same methods that people use with Captcha scripts..
Basically when the user loads the form / page.. You generate a random string/id in their PHP session and store it.. When they send the ajax requests, have your ajax check also append the string/id and require it before allowing a check to perform else return a header of 500 or something..
Using this approach with sessions, you could set a allowed limit of checks (say 5) and once the user has tried more than 5 checks, They are required to reload the page or perform a human check (eg Captcha).. Then it resets their count.. Even allow a total of say 30 within 1 hour / per IP or something.
Also use smart events to trigger when the ajax check is done, eg field/tab change or on a button press.. Or when a valid email is detected.. but say .com.au would trigger twice.
Basically this way, even if someone sniffed your JS files and tried to automate the email checker.. It would require them finding a way to append the string/id that you generate and also limit their amount of requests performed.
Beyond this, there is not to much more you can do easily.. But there are still a few other idea’s.
Most of them would work around using a PHP session / cookie.. Say for example if they check and find 3 email addresses.. Then again you set that as a limit and force them to require a manual submission or something.
See how the above suggestion goes for you, any questions do feel free to ask. But may take me a day or two to reply as weekend.. Also research how Captcha scripts work as plenty of source code for them.. As they work on the same idea.
Time Delays will simply look bad / make your site appear slow / bug the user with waiting for a response.
You need to limit the amount of look up’s per session / ip address.. Otherwise there is always a way to get past these checks.. Basically once they hit a limit.. Force the user/ip/session to wait a few minutes/hours and verify them with a Captcha script so it can not be scripted…
Javascript Security / Hiding The Source
While you can not do this truly, you can do certain things generate the JS using a PHP page with a JS header.. so <script src="https://stackoverflow.com/questions/29842122/myjscode.php"></script> and this allows PHP to check for a valid session.. So stops external requests to an extent.. But this is mostly useful for allowing JS to be only available behind a membership/login..
Multiple Checks / If Possible In This Case
Depending on your approach, is this for a user to check if they already have an account? If so.. you could combine the email check with something like their name/country/age/dob … So they would need to select two or three correct matching values before being able to get a check/response from the ajax call?
Maybe not in your case, but just thought would add this as well.