When you get right down to it, you would not be reaching out to a crisis chat or line if you did not want someone to help you with what your feeling, and if that feeling is being suicidal, then it seems to me them talking you out of it, having you call for help, or some other measure to insure your safety is likely to be the outcome of such contacts.
I can't envision someone anonymously making contact just to announce they are going to follow thru suicide, most who are dead set on suicide either don't say anything or they announce it in some fashion so that those who are part of their world know how much they hurt.
So it would seem to me tracking down someone is sort of irrelevant. But since the concern has been raised in regards to CHAT, I can put your mind at ease. The only info that is available to such a service is your IP address, and most providers use IP address pools, so IP addresses can change at any time. With some providers the pools are implemented by region or city but not always. Even if the region or city is known, it still takes a court order to get an ISP to disclose the user of that IP address at a given time. And ISP are very adamant at not disclosing such info to authorities without a court order. And court orders don't happen quickly, and reasonable cause has to be shown to get one.
Phones however are a different matter, the CALLER ID can be used to locate a person, as in most cities phone number, name, and address are part of the advanced version of 911.
I know for me being tracked down is not a concern, it seems everyone around me knows me well enough they almost know when I am having s/i and need to be hospitalized. I know my last hospitalization I was in the process of executing my plan, not saying anything to anyone. It was those who saw me earlier in the day who picked up on that something was wrong with me, and checked on my well being with very direct S/I related questions, which once I received such contacts I was committed to go to the ER or risk having unwanted drama involving police/sheriff/ems at my doorstep.