I've had credit cards for over 25 years. Never had a problem ordering online either. I've heard the horror stories and besides the ID theft ones, I think as long as you be careful who you buy from and budget your spending you'll do okay. For the last 4 years, I haven't had to leave the house to do my Xmas shopping. Everything was done 100% online, free shipping and all but a couple things came through okay. Some companies do millions of dollars of business online. I buy a lot of things from Amazon as long as it says Sold By Amazon your okay. Stay away from the 3rd party vendors on Amazon. I had a major problem with one company called AntOnline. I ordered ink cartridges thru Amazon but from this AntOnline. They sent me damaged goods and wouldn't make good. But Amazon has a guarantee policy. They made good on the 3rd party vendors screw up.
I do 95% of my business online from clothes, to toothpaste, to inkjet paper. It's all in how you can manage your money and what sites you can trust. And nowadays credit cards companies guarantee your purchases for the most part.