Pat the chicken dry and make 2–3 shallow slashes in the meatiest parts of each piece to help the marinade penetrate.
In a large bowl, whisk together the cane vinegar, calamansi juice, brown sugar, kosher salt, and ground black pepper until the sugar dissolves. Stir in the lemongrass, ginger, and 8 minced garlic cloves (reserve 2 cloves for basting).
Add the chicken to the marinade, turning to coat and working the mixture into the slashes. Cover and refrigerate 4–6 hours (or up to overnight), turning once halfway through.
Make annatto oil while the chicken marinates: Warm the neutral oil and annatto seeds in a small saucepan over low heat until the oil turns deep orange-red, 3–5 minutes. Remove from heat and let steep 10 minutes, then strain out the seeds. Stir the reserved 2 minced garlic cloves into the warm oil to bloom.
Prepare a charcoal or gas grill for medium heat with a cooler zone. Clean and oil the grates, then lightly brush with some annatto oil.
Drain the chicken from the marinade and scrape off excess solids; discard the marinade. If desired, thread pieces onto skewers for easier turning.
Grill the chicken skin side down over medium heat, 6–8 minutes until lightly charred and the fat renders, moving to a cooler zone if flare-ups occur. Flip and continue grilling 15–22 minutes more, basting with annatto oil every 3–4 minutes, until an instant-read thermometer at the thigh joint reads 165–175°F and the juices run clear.
Transfer to a platter, brush once more with annatto oil, and rest 5 minutes. Serve hot with steamed rice.