Can an unqualified property company take over the community?

No, it's illegal. According to Article 59 of the Property Management Regulations:

In violation of the provisions of this Ordinance, those who engage in property management without obtaining the qualification certificate shall be confiscated by the real estate administrative department of the local people's government at or above the county level, and shall be fined not less than 50,000 yuan but not more than 200,000 yuan; If losses are caused to the owners, they shall be liable for compensation according to law.

Whoever obtains a qualification certificate by deception shall be punished in accordance with the provisions of the first paragraph of this article, and the qualification certificate shall be revoked by the department that issued the qualification certificate.