You can buy oral contraceptives, aka "the pill" in Central America very easily. Walk into any pharmacy, or farmacia, and pick them up for about $5 US per pack. No prescription or doctor visit needed.
The pharmacy might not have the brand that was prescribed to you by your doctor at home, so be sure to make a note of the hormone levels in your pills or save a sample of the packaging, so you can see if you can match them up with another brand or a generic.
Many pharmacists in Latin America speak some English, but be prepared to ask for “contraceptivos orales” in Spanish and/or show them the labeling from your last pack.
You might find it convenient to stock up a big supply of birth control pill packs at once, but you should note that if you’ll be traveling around with your pills in your backpack, you will most likely be exposing them to temperature extremes, which can lessen their efficacy (i.e. make them stop working). Also note that doxycycline, a common malaria prophylactic, can cause oral contraceptives to fail.
So you'll want to be careful with when you get them and how you store them, but don't worry about finding them. Every city and town has at least one farmacia that will be happy to sell you the pill.
Disclaimer – I am not a doctor. I’m just a person who’s been lucky enough to have spent bunch of time in Central America and often gets asked about whether you can get bc pills there. These are just tips to point you in the right direction as you make your plans. You should never depend on medical advice from a stranger’s blog, especially if that stranger is not a doctor or nurse. Always, always consult a medical professional!