AT is used for a specific time/time frame and not intended to be divisible: at midnight. Not 11:59, not 12:01, but only 12:00
IN is used for time which is intended to be divisible: in the afternoon , in 5 hours, in March.
So, to use "at anytime" should be looked at in context. If you are going to call, you aren't going to hang up AT a certain time (meaning as soon as you call). That time you are talking or even waiting for the other person to answer is divisible. So, I would not use AT in this case.
Perhaps, a better use would be with the word "check". You can check your email at anytime. True, if you have email, you'll probably spend time replying which could be seen as divisible. If you don't though, then we are simply referring to the time as a whole (like at the end of the month). We aren't dividing that time up.