The vast majority of immigrants to the United States must have the application process started for them, by a family relation or an employer that files a petition on the immigrant's behalf. (This is sometimes, but not entirely accurately, referred to as being "sponsored.") The person or company that files the petition is called the "petitioner" and the intending immigrant is the "beneficiary." But what if either the petitioner or the beneficiary has a change of mind or the employer no longer needs the employee?
It is possible to cancel a visa petition, as discussed below. But you might want to act quickly.
To understand whether and how the petitioner or beneficiary can back out, you first have to understand a bit about the visa petition process. The initial petition must normally be filed with U.S. Citizenship and Immigration Services (USCIS), using either Form I-130 (for family immigration cases) or I-140 (for employment-based immigration cases). USCIS charges a flat fee for processing this petition. The agency ordinarily takes several months to approve or deny (see its current processing times).
After USCIS approves a petition, the immigrant, with the help of the family or employer petitioner, is brought into the process, and files an application for an immigrant visa (if coming from outside the United States via "consular processing") or for adjustment of status (if already in the United States and eligible to use the adjustment of status procedure).
If the I-130 or I-140 petition has not yet been approved when you change your mind about it, it's relatively easy to cancel. All you need to do is send a letter to USCIS, at whichever office is currently handling the petition, informing it of your decision. Enclose a copy of the receipt notice, if you received one from USCIS. That will help it trace your file.
There's a small chance you might be able to get your money back, if USCIS hasn't started processing the case, which includes cashing the check or running your credit card number. If you paid by check, try canceling it at the same time as you withdraw the application. In most cases, unfortunately, USCIS takes the position that by having taken an action on your application at all, it has basically earned the fee.
If you don't hear back from USCIS within several weeks, call its Contact Center.
If the petition has already received USCIS approval but the immigrant visa or green card has not yet been issued, you'll need to figure out which office is handling the case and send your request to withdraw to that office. In fact, they'd like you to state the reason for the withdrawal, mainly because if one party is alleging fraud by the other one, USCIS wants to note that for its records (or in the worst case, prosecute).
If, for example, the petition has been approved and the immigrant is awaiting a visa interview at an overseas consulate, the most likely office to be handling the case is the National Visa Center.
If the immigrant visa or green card has been issued, it's much harder for a petitioner to withdraw support and ask that the green card be cancelled. By this time, the immigrant is considered to have acquired U.S. status, and a petitioner would most likely have to allege fraud to have that status revoked (taken away).
You'd want to consult a lawyer who can analyze the situation based on the type of visa at issue, the risk to you if you try to withdraw it, and the surrounding circumstances.
Also note that, by this point in the process, all family petitioners have signed an "Affidavit of Support," promising to financially support the immigrant for approximately ten years after they get approved for the green card or until the beneficiary becomes a U.S. citizen, dies, or has the green card revoked. This obligation survives regardless of a divorce.
For a complex situation like this one, strongly consider hiring an immigration attorney to assist. See When Do You Need an Immigration Lawyer?.