Can the Federal government of the United States force all U.S. states to pass a mandatory vaccination mandate? Or do the states have the ability to decide whatever they want to do regardless of what the Federal government wants them to do?
Whether or not the federal government can pass its own law mandating vaccination is a separate question. The answer to the question “can the Federal government of the United States force all U.S. states to pass a mandatory vaccination mandate?” is a straightforward “No.” Such action by the federal government is contrary to the Anti-Coercion Doctrine. In South Dakota v. Dole, 483 U.S. 203, the issue was a federal law, the National Drinking Age law, which withheld 5% of federal funds (initially, 10% every year thereafter) from a state that did not have a minimum drinking age of at least 21. The Supreme Court court upheld the law not because Congress has the power to withhold federal funds in order to get what they demand from a state but because the 5% loss of funds
is not so coercive as to pass the point at which pressure turns into compulsion
In other words, pressure is legal, compulsion is not. The court did not set a dollar amount that distinguishes impermissible coercion from permissible financial incentive, but we can presume that 10% loss of related federal funds would be tolerated. In NFIB v. Sebelius, SCOTUS overturned a law that allowed Medicaid funds to be entirely taken away if a state did not change its laws with respect to Medicaid in a specific way. The court held that
The threatened loss of over 10 percent of a State’s overall budget is economic dragooning that leaves the States with no real option but to acquiesce in the Medicaid expansion... The Medicaid expansion thus violates the Constitution by threatening States with the loss of their existing Medicaid funding if they decline to comply with the expansion
There is a possible law that Congress could pass that would give states an incentive to pass a law about vaccination, but Congress cannot force states to pass any law.
The Federal government can only legislate and regulate within the powers granted to it by the Constitution
So, that means they can't "force" a state government to do anything.
However, they can pass Federal laws and, if they are within the power of the Federal government, they override state law to the extent of any inconsistency.
The Federal government has the power to regulate interstate commerce: a law mandating that any business engaged in interstate commerce (so, most banks, insurance companies, logistic companies, software companies, large retailers, large agribusinesses etc.) could only deal with customers (consumer or business) who had a "vaccine passport" is squarely within the power of the US government and, if carefully drafted, would be Constitutional. The knock-on effect is that this would affect all businesses - it's hard to run a business without a bank account.
The interstate commerce power has greatly increased the reach of the Federal government in parallel with the massive growth in interstate commerce since the second world war and, more recently with the rise of the internet. Businesses that once would have been purely local are now online and therefore are engaged in interstate commerce.
They also have the power to levy and collect taxes: a $10,000 pa tax on the unvaccinated (without medical exemption) is certainly Constitutional even if it's not politically palatable.
Of course, while the Federal government can't force a state to pass or repeal a law and may not have a head of power to allow them to intervene directly (or be politically unwilling to do so) the Federal government wields considerable influence.
For example, have you ever wondered why every single state, who normally couldn't agree that the sky was blue, has the same minimum drinking age of 21? They didn't used to and the Federal government has never "forced" them to. However, the 1984 Minimum Drinking Age Act said to the states "You can set any minimum drinking age you like. However, if it's under 21 you are ineligible for Federal road funding." Every single state "voluntarily" decided they liked Federal money more than they liked their independence on this matter.
The executive branch of the federal government can interpret existing statutes broadly and exercise prosecutorial discretion in conjunction with the interstate commerce clause to make almost any unpopular activity effectively illegal or any unpoular group of people effectively outlaws.
The legislative branch of the federal government can interpret the constitution broadly and enact laws and financial incentives to make almost any unpopular activity effectively illegal or any unpopular group of people effectively outlaws.
The judicial branch of the federal government can interpret precedents broadly and decide to legitimize acts of the federal government or delegitimize acts of state governments. This means the federal government is the ultimate legal authority to decide what it's doing is legal.
The victory of the Union over the Confederacy in the US Civil War, and the related constitutional amendments, effectively solidified the legal principle that states are not free to secede. States in the US have all agreed to be bound by federal laws (that's what being in the union means) and they can't back out.
Taken together, these observations imply that the states are totally legally beholden to the US federal government and must simply do as they are told (including appeals to the Supreme Court) or face possible military consequences. There is no reason to expect this would not apply to an executive order requiring forced vaccinations, backed by duly passed legislation in Congress and endorsed by the Supreme Court.
Naturally, there are some counterbalances that make gross abuses of power by the federal government difficult and unlikely. Congress is divided ideologically and the executive and judicial are ideologically opposed. Elected officials can be replaced periodically. Extra-legal remedies like insurrection may exist as viable alternatives to compliance with an unjust government. But, legally speaking, if the branches of federal government (and possibly enough states, say 2/3) agree, states are required to comply. There is nothing special about a vaccine mandate in this regard, except insofar as you think it affects the likelihood of rhe executive ordering it, the legislative ratifying it or the judicial upholding it.