What is the ruling on disposing of money found in the market two months ago without its owner being found, and is it permissible to donate it to charities with the intention on behalf of its owner?

This amount is subject to the rulings of a "Luqatah" (found property). Merely placing a note where it was found is not sufficient for its public announcement. Rather, it must be publicized in public places. The person who picked up the money must begin publicizing it immediately or appoint a trustworthy person to do so. Delaying this without a valid excuse is a sin and requires repentance. If there is a valid excuse, then there is no sin.

You must complete the public announcement of the found property for a full year. After that, if its owner is not found, then as a precautionary measure, it should be given as charity on their behalf. If you were not appointed, then the responsibility for the found property lies with the person who picked it up.