I don’t think there was any impropriety in Catherine’s going alone to catch the Tilneys’ in Ch. 14. After Catherine explained herself to them (Ch. 14) she returns to Putney Street on her own. Eleanor & Henry like Catherine and understand her naivety. If there had been any impropriety in her leaving their home alone I think they would have remedied it by escorting her themselves or providing her with an escort. She walks alone to different places several times in the text and there is no suggestion of impropriety from the narrator or Mr. Allen.

At the bottom of the street, however, she looked back again, and then, not at a window, but issuing from the door, she saw Miss Tilney herself. She was followed by a gentleman, whom Catherine believed to be her father, and they turned up towards Edgar’s Buildings. Catherine, in deep mortification, proceeded on her way. (Ch. 12)

In Ch. 12 Catherine walks to the pump-room and then on to Milsom Street so she can explain why she did not keep her engagement with Miss & Mr. Tilney to take a country-walk and presumably (above) returns home on her own after being told Miss Tilney is not at home.

Towards the end of the morning, however, Catherine, having occasion for some indispensable yard of ribbon which must be bought without a moment’s delay, walked out into the town, and in Bond Street overtook the second Miss Thorpe as she was loitering towards Edgar’s Buildings… (Ch. 14)

Catherine (above) walks out to shop on her own and next day (Ch. 15) Isabella sends a note asking Catherine to come to Edgar’s Buildings. There is no indication Mrs. Allen goes with her. Catherine finds out James has proposed to Isabella and later returns to Putney Street but does not inform the Allens of her news to save them the agitation she was experiencing. The next day Catherine returns to Edgar’s Buildings alone to await James’ letter with Isabella and later returns to the Allens only to find her news is no surprise to them. It might be that the reason Catherine begins walking about so much on her own is because Isabella has abandoned her in order to shower her time and attention on James. I think all this walking about takes place during daylight hours in respectable places and seem to have no impropriety attached to it. I don’t know of any definitive rules for or against a young lady walking alone so I am basing this opinion only the text of NA.