Wash and rinse pint jars and keep hot. Prepare lids according to the manufacturer.
Wash carrots well and peel if necessary; wash again after peeling.
Combine vinegar, water, sugar, and canning salt. Bring to a boil and boil gently 3 minutes.
Add carrots, return to a boil, reduce heat, and simmer until half-cooked, about 10 minutes.
Place 2 teaspoons mustard seed and 1 teaspoon celery seed in each hot pint jar.
Fill jars with hot carrots, initially leaving 1-inch headspace. Cover with hot pickling liquid and adjust final headspace to 1/2 inch.
Process pints 15 minutes at 0–1,000 ft, 20 minutes at 1,001–6,000 ft, or 25 minutes above 6,000 ft.
Cool undisturbed 12–24 hours and check seals. For best flavor, wait 3–5 days before eating.