They can fill up hospital beds though, that might not be available for you when you need one!
That said, getting vaccinated is about a thousand times easier than staying in shape. It's free and quick to get over with, whereas fitness requires sustained effort indefinitely. It's reasonable to treat people's choice to refuse something easy different from people either refusing or struggling with something difficult.
The other important distinction is that we're not gonna suddenly have a shock of people coming down with obesity and flash flooding the hospitals. The country getting bigger is not a good thing, but as a gradual thing it can be managed much more easily.
Losing weight is insanely hard getting a shot or two is easy. In fact if you had some kid who was speeding, got into a crash, and is now taking up a bed I'd argue that self control for young adults in related to reckless behavior is much more difficult to stop than not getting a vaccine.
Overweight people can't transmit their overweightness to me through a virus or bacterium that they breathe into the air.
If they could, then yeah, we'd probably have some kind of legal mandate on it. Just like we have a bunch of restrictions on smoking now, because indeed, that's something that you push into the air.
The general rule is that your rights stop where others begin. Most issues of health are purely personal, but infectious diseases are not. Because they're infectious.