There are some non-dedicated dog baskets in our list, and a lot out there. If it’s you and your pet’s first time, or you make a lot of trips, we’d recommend getting something designed for their comfort and yours. They’ll feel more relaxed, be more secure, and the weight and size will be appropriate for you. The weight and potential distraction is something that you’ll have to get used to as well.
If you do use a regular bike bucket or basket, ensure there’s some padding or a blanket to make sure your pet is comfortable, and most importantly secure their leash or harness tightly to the frame. Carriers without a strong inbuilt system fastening will only be appropriate for dogs which are very calm and will sit still. You don’t want a lack of security or a cover if your pet is likely to move around. This will be distressing for you and them, and the weight will keep shifting.