I know it's used for greetings in a restaurant or store. But what type of verb conjugation (ex: polite, plain, honorific, imperative, or something else) is it? Is it used with other verbs? Is it archaic Japanese or modern Japanese?
~ませ is the imperative form of the polite auxiliary verb
~ます which connects to the conjunctive form of verbs. According to Daijirin it's used with the verbs
召す etc (I've only seen
なさいませ used myself though, so I'm not sure how common the other ones are).
It's used in modern Japanese, and I think it's frequently used with honorific language (though maybe not restricted to it as
召す are humble language). It's used to make requests/demands more polite as well as to add politeness to greetings as in
くださいませ etc are euphonic changes from the conjunctive forms
ください (rather than being the imperative
ください), together with the imperative polite
Just to supplement the others, here is the scheme of ます：
未然形 (Irrealis form) - ませ (ましよ)
連用形 (Continuative form) - まし
連体形 (Attributive form) - ます
已然形 / 仮定形 (Realis/Hypothetical form) - ますれ
命令形 (Imperative form) - ませ (まし)
And also from nihongoresources further congujations of ます
It's polite imperative. It is derived from Edo-ben (as is all 丁寧語), but is still considered modern even though it isn't used outside 尊敬語 verbs.