Do I have to file a tax return with the IRS?

A single-member LLC does not need to file a separate tax return. The income and expenses of the LLC are reported on Schedule C, which is attached to your personal income tax Form 1040.

Also, if you are a nonresident alien, you are taxed only on your US sourced income and it appears that you do not have any US sourced income, so I would not file a US tax return.

You do need to file an Annual Report, though, with the DE Secretary of State

You should most definitively file taxes. Even though Wyoming does not have any state tax and you have foreign transactions. The US is very specific in foreign transactions. All income shall be reported to the IRS and if there is an international treaty then file the specific forms that will help you avoid paying taxes here in the US. However the fillings need to be presented if you have an active business registered with any state. Are you an american resident or citizen? If you are not then you have to file for an ITIN to present your taxes at a personal level as a single member LLC. If that is the case (that you leave it as Single Member or Disregarded LLC) you might be able to avoid "Income Tax" if the international treaty allows it. However you will not be able to avoid Self Employment Tax which is automatically created when you have a Single Member or Disregarded LLC. I would suggest looking for your LLC to be elected as S-Corp if you are a US citizen or resident. This will save you thousands of dollars.

You should definitely file a tax return with the IRS. Under U.S. tax rules, a single member LLC that is owned by a non-U.S. tax resident is required to file an annual Form 5472 and attach the form to a proforma Form 1120. The returns cannot be electronically filed, so you will need to mail or fax the return to the appropriate IRS office.

These filngs are required even if you have no U.S. source income or physical presence within the U.S. If the LLC filed an election to be taxed as a C corporation, the LLC would file a Form 1120 instead. In order to file the Form 5472, the LLC will need an EIN which you can apply for by submitting a Form SS-4. If you're a non U.S. resident without a SSN, you won't be able to apply for an EIN using the IRS online system. You need to fill out a paper Form SS-4 and mail or fax the form to the IRS. Non-U.S. residents should indicate "FOREIGN" on Line 7b which communicates to the IRS that the responsible party is a non-U.S. person without an SSN.