Why does Mrs. Bennet state that it is impossible for her and her daughters to visit Mr. Bingley if Mr. Bennet does not visit him first?

The reason Mrs. Bennet says this is because, in the Regency era, no unmarried woman must visit an unmarried gentleman at his place of residence without invitation. This is especially true if the gentleman is one who has not been previously introduced to the lady in question. While Mrs. Bennet could conceivably visit Mr. Bingley first, as Mr. Bennet cheekily suggests, the results would be disastrous.

Mrs. Bennet, as a married woman ushering her unmarried...

The reason Mrs. Bennet says this is because, in the Regency era, no unmarried woman must visit an unmarried gentleman at his place of residence without invitation. This is especially true if the gentleman is one who has not been previously introduced to the lady in question. While Mrs. Bennet could conceivably visit Mr. Bingley first, as Mr. Bennet cheekily suggests, the results would be disastrous.

Mrs. Bennet, as a married woman ushering her unmarried daughters into the home of a strange gentleman, would likely be ostracized by good society. Her actions would show a disregard for the kind of proper feminine behavior Regency society prizes. As such, Mrs. Bennet and her daughters would likely become fodder for neighborhood gossip, which in turn, would tarnish the girls' reputations. It's a tough era to live in.

A young lady's reputation was carefully protected in the Regency period, as prospective suitors were most concerned with the sexual purity of potential brides. The idea is that a wealthy or powerful man must be sure of his offspring's paternal heritage in order to ensure that no illegitimate offspring inherits his estate or title. So, sexual reputation can make or break a young woman's chance to secure a good marriage.

Additionally, rank comes into play here as well. Ideally, the person of lower rank would be introduced to the person of higher rank. In the novel, we read that Mr. Bingley's family had acquired its fortune through trade; this would make the Bingleys middle or upper middle class.

They were of a respectable family in the north of England; a circumstance more deeply impressed on their memories than that their brother's fortune and their own had been acquired by trade...Mr. Bingley inherited property to the amount of nearly a hundred thousand pounds from his father, who had intended to purchase an estate, but did not live to do it.

On the other hand, Mr. Bennet was gentry; he owned the Longbourne estate. So, according to convention and societal custom, Mr. Bingley would be introduced to the Bennet girls, after Mr. Bennet paves the way, of course.