I had to do this once for a irunguns order and just had my phone. i just used my phone browser (Brave, though if you use Chrome, Safari, etc.), you can just go in and order on the website...the taxes and shipping and stuff you pay via your credit card on the IRG page separately. It's not the greatest when the screen is so small, but it can be done.
If I recall, I paid for my order and then had a separate payment for the separate taxes or shipping or something like that. I may have gotten an email with this but I just went back into the site, logged into my account, and there will be a message that taxes and stuff due. It was two (2) transactions, both using my credit card, both using my irunguns.ca account. Make sure you create the account on irunguns.ca. I didn't the first time and it was a bit of a rigamarole ordering from the US site. You can always call them if there are issues and they can instruct you but I was in the same situation as you and I was able to use my phone browser. You'll just need to pinch zoom the screen to make sure the credit card info and shipping info etc. is all correct - I fat finger typing with my phone all the time.