Is it permissible to accept monthly aid money from a person who commits some polytheistic (shirk) and innovative (bid'ah) acts and deals with banks, knowing that the family's income source is very weak and that this money is sufficient for their obligations? And does taking this money indicate a lack of reliance on Allah?

There is no harm in accepting a gift or aid from a disbeliever, or from an innovator or a fasiq (openly defiant sinner) among Muslims, as long as it does not involve compromising any aspect of the religion or falling into disobedience. This does not contradict reliance on Allah, for the Prophet, peace and blessings be upon him, entered Mecca under the protection of Mut'im ibn 'Adi, who was a disbeliever, and Abu Bakr entered Mecca under the protection of Ibn ad-Dughunnah, who was a polytheist.