It's a an unsubstantiated claim.
During a then-current measles outbreak, on 4 February 2015 the alternative health site Health Impact News published an article claiming no one in the United States had died of the measles since 2003.
Furthermore, the article stated, more than 100 people (mostly young children) had died after receiving the MMR (measles, mumps, rubella) vaccine.
The article circulated widely during a time of increased debate over parental decisions about vaccinations, particularly among those who are opposed to the practice. In some iterations the statement was amended to specify “child deaths,” but the article itself stated there were zero deaths (among all age groups) from measles in the United States in the timeframe cited.
That claim is inaccurate, as the CDC has reported that a Washington woman who passed away in early 2015 died from (an initially undetected) measles infection:
The death of a Clallam County woman this spring  was due to an undetected measles infection that was discovered at autopsy.
The woman was most likely exposed to measles at a local medical facility during a recent outbreak in Clallam County. She was there at the same time as a person who later developed a rash and was contagious for measles. The woman had several other health conditions and was on medications that contributed to a suppressed immune system. She didn’t have some of the common symptoms of measles such as a rash, so the infection wasn’t discovered until after her death. The cause of death was pneumonia due to measles.
This tragic situation illustrates the importance of immunizing as many people as possible to provide a high level of community protection against measles. People with compromised immune systems often cannot be vaccinated against measles. Even when vaccinated, they may not have a good immune response when exposed to disease; they may be especially vulnerable to disease outbreaks. Public health officials recommend that everyone who is eligible for the measles, mumps, and rubella (MMR) vaccine get vaccinated so they can help protect themselves, their families, and the vulnerable people in their community.
Submission to VAERS alone is not evidence of vaccine-related injury or illness. The overwhelming majority of young children residing in the United States receive the MMR vaccine early in their lives, and a small number of those children have become ill (some fatally) due to reasons found to be unrelated to the vaccine. There is nothing to preclude individuals from submitting VAERS reports of sickness or death that are wholly unrelated to vaccines, and the system exists not to track substantiated incidents of vaccine injury but to identify potential trends in vaccine administration.
Few people died of measles in the U.S. between 2004 and 2015 because measles was classified as eliminated in 2000. Relatively few people in the U.S. contracted the viral infection after that, so it stands to reason far fewer would go on to die of it. And while more than 100 reports of suspected adverse reaction or death may have been reported to VAERS in the years cited, that number references unconfirmed public reports, not verified vaccine-related fatalities.
Finally, the possibility of death is not the only reason one should (or should want to) vaccinate a child against measles. As the CDC notes in their measles fact sheet, in some children measles can lead to pneumonia, lifelong brain damage, and deafness.