Is this sentence grammatically correct?

It is incorrect. It should read, "Should you have any questions or need further assistance, please contact Joe Blow or me."

One way to determine this is to remove one of the two persons you are referring to. In this instance you are referring to yourself and Joe Blow. Take out Joe blow and you get, "Should you have any questions or need further assistance, please contact myself."

That doesn't make any sense. Replace 'myself' with 'me' and you have a sentence that makes sense. Now put Joe in front of you and you have a complete, grammatically correct sentence.

Should you have any questions or need further assistance, please contact Joe Blow or me.

The first answer above is the best one, but since then many people have piped up with inappropriate "improvements", so I feel I must add my weight.

It doesn't matter if "me" *sounds stupid* or whatever argument was made against it, the point is that "me" is the right word for the object of a verb, when the subject isn't "I". I contact myself, but anyone else contacts me. And it's just a rule that the me/myself/I person comes at the end of a list.

I tend to agree with the journalist above too, that a comma between "have any questions" and "or need further assistance" wouldn't be a bad thing. (cough...although need further assistance is a verb phrase not a clause)

PLEASE don't use the word 'me'! It's totally moronic grammar! Using 'Please contact Joe Blow or I' is no use either, it sounds totally prattish! 'Either myself or Joe Blow (at such & such a number/address)' sounds better. But if this is for an advert or an article it would be better still to use your name - 'please contact Iva Notion or Joe Blow at such & such a number/address.' As for the grammar of your sentence, I've seen far worse on official documents (including ones published by schools!) and I think the most important thing is making it so that your target audience can understand the meaning, so your second option is OK!

In the event you have any questions or need any assistance, feel free to contact me or Joe Blow.

For questions and assistance, please contact me or Joe Blow.

Joe Blow and I can assist you or answer any of your questions in the event any issues arise.

Joe Blow and I will assist you with questions or further assistance, feel free to contact us.

Myself, as well as, Joe Blow can answer your questions, or assist you, if needed.

Please refer to myself or Joe Blow if you have any questions or need further assistance.

Yes, the correct form is Joe Blow or me. However, you could also insert a comma after questions, as "or need further assistance" is a subordinate clause. This means the sentence still makes sense without it.

It should read:
If you should have any questions or need further assistance you can contact Joe Blow or I at such and such a place.

Maybe: Should you have any concerns or assistance, please contact myself or Joe Blow.

Here's a good book that will help clear all that up.

Woe is I (Patricia T. O'Conner)

It looks ok to me. My only suggestion would be to remove the comma and insert the word "then."

No, it should read:

Should you have any questions or need further assistance, please contact Joe Blow or me.