No, you are absolutely NOT more likely to have a SD because you are at home. In fact (whilst no emergency situation can ever be completely ruled out, wherever you give birth) you are probably less likely to experience this at home, as you have far more freedom of movement.
At home, you are unlikely to be giving birth lying on your back on a bed. It is this position that is a major contributory factor in SD, as it reduces the pelvic measurements, so the baby's shoulder can get stuck behind your pelvic bone. If you're lying down, your sacrum/coccyx cannot move back and out to enlarge the pelvic space and let the baby move through as it should. This is especially important if you have a raised BMI.
The maneouvres that free a shoulder dystocia in the vast majority of cases are simply ways of opening up the pelvic diameters (getting you onto all fours, or doing something called McRoberts which is pulling your knees right up by your ears). But if you're not lying on a bed in the first place this is far less of an issue, iyswim?
These manoeuvres are exactly the same at home as they would be in hospital, and all midwives are trained and updated yearly in performing them. (Re: doing an emergency c-section - the head is already out, for a start, so doing a section would involve replacing the head into the pelvis, a manoeuvre called the Zavanelli. I really cannot emphasise enough how incredibly rare and unlikely that would be - virtually all cases of SD are resolved with McRoberts or all-fours positions.)
Please do try not to worry - it really is unlikely. Have a lovely birth.