Do you think it is okay if I take pictures of my coworker's messages and anonymously send them to hr?
What did they say in their messages that makes you want to send them to HR?
She is texting her clients and flirting and I'm afraid she will take my clients away.
It's inappropriate for you to be spying on your coworker and their messages with clients, because that's none of your business. Also, you definitely shouldn't send those messages to HR. I suggest that you forget about what your coworker is doing and focus on yourself.
Screw this I have a plan and I'm going to follow it.
It sounds like there is some competition at work and you should try to find more appropriate ways to keep your clients. Use this as a way to motivate yourself to be creative as you approach your customers.