What is the grammar of "用餐曅請隨手整理桌面 座位有限請勿休憩估用"?
This is probably easier with some punctuations:

用餐畢， 請隨手整理桌面。

座位有限， 請勿休憩佔用。

Let's break it down:

用餐畢: 用餐 means 'to dine' or literally 'to use meal'； 畢， short for 完畢, means 'to finish, to complete' (intransitive sense only); so altogether - 'after your meal'.

請隨手整理桌面: 請 - 'please'; 隨手 means 'while at it, while doing it' or literally 'to follow hand'; 整理 - 'to arrange, to put in order', in this case 'to tidy up'; 桌面 - 'tabletop' ; so altogether - 'please readily tidy up the table'

座位有限: 座位 - 'seat'; 有限, means 'having a limit, limited' ; altogether - limited seating

請勿休憩佔用: 請 - 'please'; 勿 - 'don't, not' ; 休憩 - 'to rest' ; 佔用 - 'to occupy'; altogether - please don't rest (here) and occupy (the seats)

The whole thing can be translated as:

Please tidy up the table after your meal; limited seating, so please don't rest here and/or occupy the seats.

Personally I'd have used 清理 instead of 整理, as 清理 implies cleaning and taking your mess with you whereas 整理 usually means to put things in order. It just sounds a little unidiomatic. Maybe it's more idiomatic in Taiwan (I assume this is where you saw the phrase from)?