Armenia is one big campground. There are virtually no fenced off fields and people are very hospitable, so although you should be respectful and conscious of private property, you have a great deal of freedom to set up overnight camp in most beautiful spots you come across.
Despite all of the great camping opportunities, most Armenians don't go camping and there are few organized campgrounds. Lake Sevan peninsula in the summer, known as "kghzi" to most Armenians is the big exception. The shores and forest there are lined with many car campers, who are barbequing, picnicking, drinking and toasting.
There are a few shops that sell tents, but you may prefer to bring your own gear. You can also contact a couple of travel agencies that can organize camping tours. Avarayr Travel, Asbar Travel and DA Travel can all create tours to suite your needs, or provide suggested itineraries. Asking if they'll rent you some gear is also an option.