I expect the company wanted to pay you for a product (on a purchase order) rather than as a contract laborer. Whatever.
Would they be willing to re-issue the check to you as a sole proprietor of a business named ABC Consulting (or anything like that)? You can register your sole proprietor business with the state using a "Doing Business As" (DBA, or fictitious name), and then open the bank account for your business using the check provided by the customer as the first deposit. (There is likely a smaller registration fee for the DBA.)
If they won't re-issue the check and you have to go the LLC route...
Scrounge up $125 doing odd jobs or borrowing from a friend or parents. Seriously, anyone can earn that amount of money in a week or two.
Besides the filing fee for the LLC, your bank may require you to provide an Operating Agreement (which is not required by the State). The Operating Agreement can be simple, or more complex if you have a partner (even if it's a spouse). If you do have a partner, it is essential to have such an agreement because it would specify the responsibilities and benefits allocated to each partner, particularly in the event of equity distributions (taking money out of the business, or liquidating and ending the LLC).
There are websites that will provide you a boilerplate form for Operating Agreements. But if your business is anything more than just single member LLC, you should pay an attorney to draw one up for you so the wording is right. It's a safeguard against potential future lawsuits.
And, while we're at it, don't forget to obtain a EIN (equivalent to a SSN) from the IRS for your LLC. There's no cost, but you'll have to have it to file taxes as a business for every year the LLC exists and has income.