Gusto kong kumain ng kaymito, pero walang kaymito dito sa America, kaya kailangan kong pumunta sa Pilipinas.

The sentence is grammatically correct, however, it would not sound realistic if we are to consider its actual meaning.   This is what it translates to:  I want to eat kaymito (now), but there is none here in America, that’s why I need to go to the Philippines.

To say “gusto kong kumain” indicates urgency, hence, I inserted the “now” to convey that meaning in English.  It would, therefore, not be appropriate to say “gusto kong kumain” unless the food you are referring to is easily obtainable in sense of time and space.  If going to the Philippines were as simple and quick as just crossing the street, then the sentence would be all right.  However, that not being the case, the reality of the situation indicated in the second part of the sentence will not agree with the urgency expressed in the first part.

BTW, you will usually see “caimito” (Spanish) as how the word is spelled here.  At times “kaimito” is used, and this is actually the first time I have encountered it spelled with a “y”.

These are possible alterations you can make for the sake of coherence:

Gusto ko ANG kaimito, pero walang kaimito dito sa America, kaya kailangan ko PANG pumunta sa Pilipinas para makakain nito. = I like THE kaimito, but..., that’s why I STILL have to go to the Philippines to be able to eat it/this.

Mahilig ako sa kaimito, pero .... para makakain nito. = I am fond of (I like the) kaimito(s), but...  to be able to eat it/this.