Can a mother who is a citizen of the us file for her daughter and children within 6 months?

The Detail is confusing, and I don't think is grammatically correct.

An approval could happen within that time frame but visa availability will definitely take some years depending on what preferences they fall into.

She can apply for her daughter, but it will take many years.

The mother can file immigrant petitions for her children any time, even tomorrow. The petitions' approval will take about a year. After the petitions are approved, the children will have to wait for their priority date: only 23400 visas are given every year for unmarried children of U.S. citizens, and 23400 visas per year - for married children of U.S. citizens; so, the children of the person in your case will have to wait until all the people who got their immigrant petitions filed (and approved) before them receive their visas; right now, the wait time is 7 years for unmarried children (19 years for citizens of Mexico, 15 years for citizens of the Philippines) and 10 years for married children (19 years for Mexicans, 20 years for Filipinos) It makes sense to file the petitions, anyway. First, because neither the waiting times nor the filing fees are likely to get any smaller. Second, the waiting time does not start until you file the petitions. And, most importantly, your grandchildren must be under 18 when the petitions are filed; otherwise, they will not receive visas under their parents' approved petitions.

I am not sure that I understand your question. A US citizen may file an I-130 immigrant visa for their children at any time. Grandchildren would not be included in that.

Yes, a mother who becomes a US citizen may apply for children (and their immediate spouse and children) who are over 21 years of age.

That filing of a petition can take place on the day she becomes a US citizen. That is not the issue. The issue will be that if her daughter is over the age of 21, it will take about 5-6 years before the daughter and grandkids will be eligible for an immigrant visa on the basis of that petition.

No. The adult child of a US citizen is an FB-1 and the current visa priority date for that preference classification is August 2005. If the daughter is married, she is an FB-3 and the current visa priority date is May 2002.

Married sons and daughters of US citizens have to wait at least 10 years before they can apply for an immigrant visa or adjust their status in the US. Their children will also be allowed to immigrate with them as long as their children are under the age of 21. Unmarried sons and daughters have to wait at least 8 years.