HTS is basically an Emitter Locating System hooked up to a targeting computer that is interfaced with the aircraft. Without HTS - or a similar system - the aircraft has to rely on either its own radar warning receiver (for HARM SP mode) or for HARM's seeker to pick up a target itself - which is of course far shorter than a dedicated ELS could pick it up at.
Basically, they can still fire HARM without HTS, but only at shorter range or in pre-briefed mode against a known target (area). For pre-briefed mode, they pretty much need a ELINT/ELS aircraft such as a RC-135 Rivet Joint (or a USN EP-3E) in the area to pick up a target at long range for them, and supply them with a target area to fire the HARM into. Rivet Joint is NOT accurate at range though, hence why a rangefinding ELS such as HTS makes the system far more accurate. HARM carriers without HTS will also be far less accurate even in SP or TOO mode.
When firing without HTS, the launching aircraft basically only has a bearing (provided by onboard RWR or by Rivet Joint) towards which the HARM is fired. HARM then flies into that direction and hopes to pick up the hostile radar on its own seeker.
Afaik, all F-15 in current USAF inventory have been given the necessary software upgrades to carry HARM, but HTS has never been integrated on the system. Supposedly the USAF currently has around 550 HTS pods for its 1300 F-16, with 200 pods being of the latest generation (R6/R7).
TAS provides a similar capability to HTS for Navy aircraft, but is not a podded system - but embedded in two underwing pylons (on all F/A-18C/D and iirc EA-6B and EA-18G). HTS and TAS themselves can track less targets in less field-of-view than the old APR-38 of the F-4G Wild Weasel, but are faster and more accurate due to software improvements. There has been a somewhat similar system tested with the F-15, intended for a (cancelled) dedicated SEAD variant (F-15/HARM or F-15/PDF).