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Yes, there is another meaning other than 'say'.
It is one of the three typical humble expressions:
お/ご + verb/noun + する
お/ご + verb/noun + いたす
お/ご + verb/noun + 申し上げる
And, of course, you can append 'ます' at the end of each of them to express politeness to the listener.
That is, お待ち申し上げます is a humble and polite way of saying 待つ.
お + 待ち (a conjugated form of 待つ) + 申し上げ (a conjugated form of 申し上げる) + ます
Just for your information, usually pure Japanese words are preceded by お, while words of Chinese origin are preceded by ご.
