Sure they can get married quickly. They can go to their parish priest, his or hers. Or they can invite a priest to their home, his or hers. Or they can go to a justice of the peace. Bring a friend for witness or ask the justice to ask his wife, or clerk or neighbor.
I am working on indexing old marriage records from this time period and earlier. Many of these couples were married in his or her home, with parents witnessing, or a friend, and the minister present. Or they were in a town hall or other town office. Quick and easy. What I noticed as I got further back in time is that they could get married under 'the banns' or a license. If they were young, they needed parents' permission.
I believe many of my ancestors did this and several of them had an 'early baby' seven-six or even five months later. I do believe it was 'no big deal.'
I just want to add something about the 'early baby thing.' I once asked my grandmother, born 1903, middle-class, old New England Yankee family, if being pregnant at time of marriage was so 'horrible' or if people would gossip, etc. (She had one of these early babies herself.) She laughed and said so many people did it, it would be hard to find a family who didn't have a cousin, sister, aunt, or someone in the same position. She said, yes, people might gossip a bit, but the woman you're telling might have had the same thing happen to her. It was common, in other words, probably as much as it was today and according to my grandmother, nothing to be shameful about.
(One more thing: there certainly was a stigma attached to an unwed mother, but a woman who entered marriage already pregnant became 'respectable' upon that marriage. In the society my grandmother lived in, the boy or young man, if reluctant, would be pressured to marry the girl by family, friends, his church, etc.)