Absolutely, and most experienced business travelers maximize this opportunity aggressively. As long as your company policy doesn't explicitly prohibit it, earning airline miles, hotel points, and credit card rewards on employer-funded travel is standard practice and expected.
Attach your loyalty numbers to all bookings automatically, avoid fare classes that don't earn miles, and consider accommodations based on your status progression goals. If you ask Otto The Agent, it can provide calculations like: "This United flight earns 1,250 Premier qualifying miles versus 800 miles on the American option."
Credit card optimization adds another layer. Know which cards earn bonus categories: "Use your Amex Gold for this restaurant charge (4x points) and Chase Sapphire for the hotel (3x points)." These category bonuses compound significantly over a travel year.
The holy grail is combining company-paid travel with personal point earning and credit card bonuses to fund future personal vacations. I've earned enough points through business travel to take my family to Europe annually - essentially turning my company's travel budget into personal vacation funding.