No, neither the word "veterinarian" nor the shortened version, "vet" should be capitalized (capitalised) in modern English. By "capitalized" I mean the first letter of the word is in upper case. The first letter of words should be capitalized under certain rules. These rules have evolved over the years. English has become much less formal. Accordingly, the rule on capitalizing the first letter has become less demanding.
Places and geographic terms are started with an upper case letter. "New York" is one example. "London" is another and "North America" is a typical example.
The word "veterinarian" describes a profession, no more. You wouldn't call a train driver a "Train Driver", would you? The fact that a veterinarian has a better status (some would say) in society does not make any difference.
Two last points. The word "vet" can also mean a person who has retired from the armed forces (short for "veteran"). It is better therefore to spell out the entire word on the internet so that Google can understand its meaning.
The word "internet" can be spelled with a capitalized first letter - "Internet". But modern English, particularly internet English favours ease and speed hence lower case.
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