Most online stores will be using postal or courier services to deliver the items to you. With millions of gifts and hundreds of millions of Christmas cards being delivered around the world, these courier delivery services may not stick to delivery schedule during the festive season. Quoted courier shipping times and courier parcel delivery schedule mentioned on the store website(s) you are purchasing from should only be taken as a guide, and if in any doubts, take the express delivery option to ship the gifts delivered to you.
Online stores are no different from your local store when it comes to having items in stock. Gifts purchases can reach frantic levels the closer you get to the big day, and the most popular items are just disappear from the estores for being sold out.
Security, privacy, and reputation are all the standard issues one should be concerned with, but when purchasing gifts online. Give a thorough reading and understand the online store's returns/exchange policy. We have all seen Christmas gifts that didn't quite fit properly, or found out that someone else had given the same gift, or that the person already owns the gift we had given. Most reputable websites will have their returns/exchange policy clearly posted and accessible from their homepage. If they do not, you should contact them about their policy prior doing a purchase.
Do not be afraid to ask questions or advice from an online merchant. If you phone them, email them, or use their live chat facility, not only should they be able to give you advice, but their response will help you determine what their level of service is like. In fact, even if you are happy with their website and their products, and have no real questions, ask one anyway and see how long they take to respond!