If this is a foreign based company with no locations or business in the United States there are no requirements for that company to report or file anything with Us tax authorities.
So you should not expect that company to collect or file any IRS forms.
You will need to maintain your own records, since as a US citizen or resident you are required to report all of your world wide income (regardless of what the company does or does not file or report).
Please ask if you need more discussion or clarification.
Ok, so what types of records do I need to be keeping, and is there a particular way to keep them? Obviously, I'm new to this.
There is no required format but you just need to keep track of your income and business expenses.
You can do this by hand, in a spreadsheet, with a software program like Quicken, an online service like GoDaddy Bookkeeping, and so on. Whatever allows you to accurately be able to report your net profit which is your income less your business expenses can be used.
For more information on recordkeeping see
Hope this helps in your new endeavor.
If you need more help or have another question, please put my name in the question to allow me to assist.