I typed out a detailed reply, pressed the "Post Response" button and somehow the forum lost my post. I'll briefly share the key points:
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"Vet Las Vegas" seems like a perfect keyword for you. I would not suggest targeting "Las Vegas veterinarians". No one will type that phrase into google when performing a search. Google will relate other phrases to that phrase of course, but why take the indirect route when you can directly target more relevant key phrases. I would try even minor tweaks like "vets Las Vegas", "Las Vegas vet", or even "pet dr Las Vegas" before I thought of "Las Vegas veterinarians".
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Your site title has the keyword "Vet Las Vegas". The keyword is not used on your home page even once. You use "Pet Medical Center of Las Vegas" six times. You rank #1 for the term. It appears you have plenty of opportunities to work in better phrases.
How about change ""Keep In Touch: Be the first to learn about news, tips, and special offers from Pet Medical Center of Las Vegas." Possibly change "Keep In Touch" to "Are you looking for a Las Vegas Vet?".
Also "Providing Excellence In Veterinary Care For Dogs and Cats Since 1993." could be changed to "Providing Excellence in Las Vegas Vet care for Dogs and Cats since 1993".
You asked about links. You only have 4 links from 2 domains. The world is open to you. Local directories, hotels that allow pets, police departments, pet groomers, etc. Anyone and everyone who both has a web site and may direct people to your office should be a target.
A sign in your lobby or exam rooms would be great. When I take my cat to the vet, I wind up reading the labels on specialized pet food because there is nothing else to do. How about asking your satisfied customers for a link? A simple sign asking others to link to your site can help.