Yes, this is normal. When I applied for my company card, I have to submit to the credit card company. However, it does not work like a normal credit card. The credit card is almost always given to you regardless of your credit score. They only check if you been black list which can happen if you were previously delinquent on your debts within the same company. Opening a company card does not get reported on your credit history that you opened a new credit line, and as you pay the balance, your credit score does not get a bump (I think a very slight one, if that when I asked about it to the credit card company). You cannot use the card outside of company expense. You might get away from time to time, but most of the time you cannot carry a balance over, or if you do it gets reported to the company and they cancel your card.
The only way it gets reported on your credit history is if you don't pay it or pay it late. I believe this is why companies do it this way so if you refuse to pay, it gets on your credit history, and on top of that it saves the company money by eliminating an entire department dedicated to paying cards. Typically you submit expense reports with receipts, then the company cuts you a check, and then you pay the balance.
My advice is to keep track of the payment date ranges. That way as you get back from a trip, you can easily wait for the check to come in the mail and you can pay your balance. Otherwise you might have to pay out of pocket until the check comes.
Outside of this, you do not have to use the card, if you don't want to. You can simply keep it for large expense items like hotels, car rentals, gas, or plane tickets. But small stuff you can usually pay out of pocket during your trip and get reimbursed. Generally speaking, there's no way they would know if you paid by your card or not. Once you get back from the trip, you submit your expense receipts and get reimbursed. Keep all your tickets and receipts and tolls you get and be sure to submit and pay in a timely manner.