I quote from B. Elizabeth Pryce's excellent book "How Good Is Your English?" (1985) :-
Countable nouns can carry indications of large or small numbers by the use of words few, a few, many.. Few means a very small amount where more might have been expected. A few means only a small number where hardly any might be expected. Many means a good number, more than several.There have been few oranges in the trees this year, in fact a very disappointing crop.
There were still a few oranges left on the trees after they had all been picked for marketing.
There are many oranges on the trees this year, promising a good crop.
So. The answer to your question appears to be that it is grammatically correct to "invite many people" to a party.
'A lot of' is to be used in positive sentences such as: They invited a lot of people to the party.
'Many' is to be used in negative sentences and questions such as: They didn't invite many people do the party. Did you invite many people to the party?
My question to you all out there is; should I be correcting students who say, 'They invited many people to the party'?
I don't think I should but the course book begs to differ.
I asked for some help with a grammar qestion a while ago and I received great feedback, thankyou.