First of all, this is probably not legal, either because it is in substance an effort to avoid immigration laws, because the workers are inaccurately classified as independent contractors when they should be classified as employees, or because it amounts to money laundering. It also would be a violation of immigration laws to the extent that someone knew, or was willfully blind to fact that the person providing the services was not authorized to work and was aiding and abetting an immigration law violation.
An LLC is also a poor mechanism to use to do something like that. Among other things, the application for the EIN would reveal facts about the underlying single member that would be available for authorities to summarily prove their case of an immigration violation which would also be flagged for audit on tax returns.
A state law corporation taxed as a C corporation could be potentially more effective, although not quite in the scenario laid out in the original post, and are used sometimes in the construction industry. There is no obligation to issue a 1099 to a state law corporation or to get its taxpayer ID (unless it is practicing law or in a few other exception cases) when making payments to it, which is the practical trigger for needing a taxpayer ID on Form W-9 from independent contractors in most cases.
Also, so long as the C corporation is incorporate in a U.S. jurisdiction and does not pay dividends, it does not have to report information about its shareholders to the IRS in most instances, and while shareholder/officer/director information including that person's SSN must be used to obtain the initial EIN of the C corporation, the EIN of the C corporation does not change if its all of its shares are sold to someone new, and it has new officers and/or directors. Furthermore, non-citizens who are not authorized to work in the U.S. are not prohibited from owning shares as passive investors in a U.S. C corporation, or from receiving dividends from it, so long as there is tax withholding when the dividends are paid by the C corporation to its shareholders.
So, if a manufacturer or wholesaler enters into a distributorship contract with a C corporation to sell knives on a commission basis, for example, and the C corporation does so, and then pays tax, first at the C corporation entity level and then withholds tax on dividends paid to its shareholders who are not U.S. citizens, it has not, on its face, obviously violated any immigration laws or tax reporting laws, and has not in any officially filed document confessed to violating the law.
The problem with that arrangement is that if the reason that the C corporation is able to generate revenues is that someone who is not authorized to work in the U.S. is doing the work, and is benefiting from it economically, for example, in a single shareholder C corporation, then the person doing the work and the C corporation are both engaged in immigration law violations and tax law violations as well because the payments characterized as dividends would actually be deemed to be wages from which wage withholding was not made (which someone who is aware of the situation could tip off ICE or the IRS about).
If the manufacturer or wholesale distributor knew about this arrangement, or was willfully blind to it, that manufacturer or wholesale distributor could have liability or face criminal charges for conspiring to commit immigration and/or tax law violations.
But, if the services were provided by a C corporation in a context where entities of all manner routinely provide services (e.g. in construction projects), the company hiring the C corporation to perform the services might legitimately avoid liability for not trying to figure out if the C corporation was operating legally with respect to its internal affairs or not, and would not have liability for not reporting payments made to the C corporation because payments made to C corporations (subject to exceptions for law firms and a few other circumstances) do not have to be reported on form 1099.
Historically, the "double taxation" of C corporations at both the entity level when profits are made and the shareholder level when dividends are paid, had made C corporations unattractive and effectively imposed a tax penalty on trying to use this approach to avoid other laws. But, C corporation tax rates and tax rates on "qualified dividends" (which most C corporation dividends to long term shareholders are), are both currently so low that the double taxation burden is quite tolerable. Moreover, if the non-citizen shareholders in a C corporation are not residents of the United States, in general, dividend payments to them are not subject to U.S. income taxation.
One could imagine, for example, an undocumented non-citizen in the U.S. or a non-citizen whose visa did not allow him to work, creating a C corporation of this type with his or her non-citizen relatives abroad as owners, working in the corporation without pay while, for example, being a student with scholarship support as a primary occupation on a student visa, paying tax on the profits at the corporate level, and then making dividend payments to the corporation's shareholders abroad tax free as a form of remittances to family members that the student is trying to support. Since the C corporation would not get a compensation deduction for the work done by the person whose immigration status didn't allow them to work, it wouldn't even be outright tax evasion. But, this would still probably violate immigration laws prohibiting work by undocumented non-citizens and by non-citizens with visas that don't authorized employment. Also, substance over form doctrines could be used to recharacterize the dividend payments as wages to the working generating the corporation's income followed by a gift from that worker to their family abroad.
One should not rely on this answer, however, to assume that you are in the clear, as there are many form over substance doctrines in both tax law and immigration law, and this involves highly technical areas of the law in which it is easy to run afoul of regulatory exceptions and details not mentioned in the answer.