Is the sentence grammatically correct and is "get" or "have" correct?

Your sentence is rather long and unwieldy. You'd be better off breaking it into more than one sentence:

I have completed the DP training program. I will be able to submit my documents for verification to Nautical Institute in UK for issuing the Unlimited DP License once I get the original letter.

I would further rework that second part thusly:

Once I get the original letter, I will be able to submit my documents for verification to Nautical Institute in UK for issuing the Unlimited DP License.

I am an American so this might be different in the UK, but over here, we refer to it as the UK rather than just UK. Same for The Nautical Institute but, as I am unfamiliar with that organization, I can't say for sure that it should have an article. However, if it's this Nautical Institute, then it definitely needs a "the" in front of it.

So, guessing what at what you mean, I would suggest:

I have completed the DP training program. Once I get the original letter, I will be able to submit my documents for verification to The Nautical Institute in the UK and they will issue the Unlimited DP License.

As to "get" versus "have", "get" implies that you do not yet have the original letter whereas "have" means that the original letter is in your possession already. Either is grammatically correct.

Edit: I wasn't quite clear on the last point. "Once I have..." does refer to the future point at which you have it in hand whereas "Once I get..." is referring to the action of receiving. "Have" is just slightly after "get" but either is fine.