As a result, Bob becomes ill and dies. Could Eve be tried for Murder,
Manslaughter, or some other crime, as she chose not to be vaccinated
against a disease that she (in-directly) passed on to Bob and killed
There are basically two distinct issues here. What is the duty? And if a duty was breached, what intent is necessary to breach it?
There is not a legal duty to be vaccinated. There is a duty to use reasonable care not to hurt others. The duty not to hurt others could be satisfied by not seeing Bob in person, by wearing a mask around Bob or by having other non-transmission means available, in addition to being vaccinated. But Eve didn't do any of these things.
We don't know if Eve had any reason to think that she presented a risk of infection to Bob because she could have passed the virus to him while she was asymptomatic.
We also know, by the assumption of the question, that Eve was the source of the infection. But, in real life, proving the source of an infection beyond a reasonable doubt is very challenging or impossible. This must be established for any homicide crime.
There is no indication that Eve knew she was transmitting the virus to Bob, or that Eve intended to transmit the virus to Bob (if she intentionally spat in Bob's face intending to infect him that would be a different matter). At most, her conduct was reckless, but if she was asymptotic and has no idea that she was doing something that was actually putting Bob at risk, her intent could be as slight as negligent (for tort law purposes only) or criminally grossly negligent.
Since she lacked the necessary intent to commit murder (i.e. either an intention to kill, or an intention to inflict grievous bodily harm), she could not be guilty of the offense of murder.
There are three types of voluntary manslaughter in England, none of which apply here: "There are three types of voluntary manslaughter: that resulting from loss of self-control; that resulting from statutorily defined diminished responsibility; and killing in perseverance of a suicide pact."
So, this leaves involuntary manslaughter as the most serious possible homicide offense. Involuntary manslaughter could encompass either reckless conduct (i.e. "the unlawful act must be such that all sober and reasonable people would inevitably recognise it as an act which must subject the other person to at least the risk of some harm resulting therefrom albeit not serious harm") and is usually in furtherance of some other criminal offense, or in the case of "gross negligence manslaughter", negligent conduct that is a far greater level of wrongdoing than the negligence that would suffice for civil tort liability.
Gross negligence manslaughter is the most plausible charge and is itself a hard call that involves judgement and discretion on the part of the trier of fact (i.e. the judge in a bench trial, and the jury in a jury trial) that is exercised on a case by case basis considering all of the circumstances. Also, to be clear, the wrongful act in a gross negligence involuntary manslaughter case would be transmitting the virus (which could have been prevented multiple ways) and not failing to get vaccinated itself.