I don't know if it's "appropriate" or not, but it's not a huge deal. People who work in the field, especially at the OB's office know women have kids (obviously, or they'd be out of a job!) I'm pregnant with my fourth baby and often have to take my first three with me to appointments. They are 5 years old, almost 3 years old, and 13 months. It's either that, or don't go, so I take them. However, if I know I'm having an internal exam, indepth u/s, etc. I will do everything within my power to arrange babysitting or reschedule.
I'd say if your son is 10 months, you probably won't even need an exam if you had a post partum check up. You'll probably show up, talk to the dr. and they'll prescribe you the birth control, so don't stress too much. If your appointment was hard to get, keep it and take your son.
With my kids, it helps to take snacks and a favorite activity. My kids usually go in with me and sit in the chair with their snack while I speak to my midwife or get examed if necessary, and they're usually well behaved. If your baby is 10 months, he may be fine in his stroller, just while you're in the actual exam room. Give him his favorite snacks he may not normally have (I don't know how much your 10 month old son eats) but I'm always successful with cut up strawberries, grapes, goldfish, etc. If your son is walking, let him walk in the waiting room so he can get his "wiggles" out, and will likely be more patient in the exam room.
It may not be ideal, but doing things with kids in tow and making it work is part of being a parent! Just wait a few more months and you need to go to the DMV, get important shopping done, etc, and your toddler has a melt-down! Don't stress too much, be prepared to entertain him, have appropriate expectations, and a sense of humor. You'll be fine!
mama of three, pregnant with four!