To answer the question as asked:
You'll have a Shimano Nexus gear hub, so you need a sprocket compatible with that. (Any reasonably competent mechanic should have been able to tell you that!)
Sheldon Brown has a page on Nexus and Affine geared hubs which includes info on sprockets: https://www.sheldonbrown.com/nexus-mech.html.
You should be able to find Nexus sprockets at most major online bike component retailers. E.g., at Universal Cycles: https://www.universalcycles.com/shopping/product_details.php?id=4171.
To be sure about what you are buying, I think you will have to get the wheel off to look at the existing sprocket to compare with images of new sprockets at inline retailer pages.
Don't be too scared of getting the wheel off, it's a straightforward process. Park Tool has a great YouTube video detailing the process of replacing a single speed chain which includes removing and re-installing the wheel:
I believe there are no specialized tools needed to get the wheel off or to replace the sprocket, beyond wrenches, screwdrivers, pliers etc. You will need a chain tool if you are replacing the chain. See the Park tool video.
There are a few videos on YouTube that show how to replace the sprocket. I found this one:
Now to address 'implied' questions:
You may not need to replace the chain or sprocket, you do however need to give it a good clean and lube! You may find that cleaning and lubing it ameliorates any problems you are having with it.
There are several methods of determining if a chain is worn out. Chains appear to 'stretch' as they wear out (they do not literally stretch, they get slightly longer as they wear because the holes in the links the pins go through get slightly bigger, allowing the individual links to move slightly further apart). You can find answers on here that address determining chain wear by measuring the degree of stretch. You can also find answers about determining sprocket wear.