TL;DR: No, Congress cannot. However, this can be mandated at the State level.
So, I found an article discussing this (though it's related to the SARS outbreak). This mostly discusses whether doing so violates the 14th amendment to the U.S. Constitution. Here is a relevant part (of the earlier article):
When determining the legality of a statute enacted to protect public health and safety, the Court found it immaterial that a portion of the medical community thought the vaccination worthless or even injurious. The state has the right to choose between opposing medical theories and to refer the matter to a board composed of persons residing in the affected location who are qualified to make a determination. The courts do not become involved in legislation formed under the state’s police power as long as it relates substantially to public health, morals, or safety and is not a plain, palpable invasion of rights secured by fundamental law 5. Furthermore, it is immaterial whether or not the vaccine is actually effective, so long as it is the belief of state authorities that the mandatory vaccine will promote common welfare and is a reasonable and proper exercise of the police power . It is of paramount necessity that a community have the right to protect itself from an epidemic of disease which threatens the safety of its members.
So it appears that the courts have ruled that mandatory vaccine law is okay under the 14th Amendment. Though, a nuance here is that it seems to be the states doing it here, not the Federal Government.
As far as whether the Federal Government specifically could do this, that doesn't seem likely. They'd likely be fighting the 10th Amendment to the Constitution:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
And since this power isn't granted explicitly to the Federal Government (that I'm aware of) anywhere in the U. S. Constitution, it is implied to be reserved for the states.
The one angle the Federal Government might have is the Commerce Clause. But so far, it seems likely the Supreme Court would rule against this (emphasis mine):
That the states are on firm constitutional ground in repealing personal belief or religious exemptions doesn’t mean the federal government can abolish such exemptions. In its 2012 ruling on the Affordable Care Act’s individual mandate, the U.S. Supreme Court said that the federal government’s authority over interstate commerce does not permit it to compel individual action. If Congress can’t require individuals to have health insurance, or eat broccoli, it can’t mandate vaccination either.
So no, it doesn't appear Congress can. But there is precedent that individual states can.