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The answer depends on wether you need caching or not. Keep in mind that ajax requests are sent by the browser and therefore don’t cost you any server resources. Caching is only really necessary if the third party api you are using can’t handle a huge amount of requests.
If you’ve decided you need caching, you’ll have to access the api via your backend, in your case that means using php. Of course you could also write your own api dispatcher / cache in something like NodeJS and use this as a microservice, but that sounds overly complicated for a small project.
In my opinion you are best off to just access the api via ajax in vue, it’ll save resources and will be the easiest way to go, everything else seems just redundant.
Testing a third party api can be tricky and in your case is probably redundant. What you rather want to test is how your application integrates with the api. You also probably want to write a mock for that api so that you can run your tests without depending on the api.