To legally search a car, you first have to either witness an accident or justifiably pull over the car (No turn signal, running through a red light, expired registration, etc).
Then, you look for criminal activity (DUI, no license, fake insurance, etc.) or certain indicators (alcohol odor, traces of green crumbs, white powder under the nose, etc.)
The handbook lists these indicators in detail.
Searching and inventorying

In real life, police (at least in the US) can legally perform a type of search called an inventory search; these searches are usually for vehicles that have no insurance, have significant damage, have expired registration, have broken lights, have smoking tailpipes, have loud engines, the owner has a suspended/revoked license, etc. They can do this as long as there is a clear policy on how the inventory is done, how the items are listed, and how parts are listed. Police can not search a car for evidence under the guise of an inventory search (this relies on discretion much of the time).
In PSPO, you can not inventory a vehicle right away just based on those things alone. You have to look for indicators such as red eyes, alcohol odor, or green crumbs. When you have those indicators, you can inventory the car after you've searched the driver. If any illegal items or unregistered guns are found, you can justifiably arrest the suspect. Either way, tow the car afterwards.
Inventorying a car

If a suspect pulled over for a traffic violation, such as speeding or running a red light, has a clean vehicle record (valid insurance, valid license, and valid registration), you can still search them and their car if alcohol or drug indicators show up. However, it is recommended to only search the driver's side and passengers' side doors (unless the trunk has a visible window or the bed of a pickup truck is open).
Under the Supreme Court in Arizona v. Gant, searching a non-impounded suspect's vehicle is only valid if it's done to areas within the suspect's immediate reach.
You can also exercise discretion when it comes to expired insurance, expired licenses, and expired registration that is recent. In that case, only search the doors (and/or the open beds of pickup trucks and trunks with actual windows).

Sometimes, a suspect may flee during a vehicle search or an inventory search. If that happens, chase after them and aim/use the taser. Then, arrest the suspect for fleeing and continue the search
If a suspect gives off indicators that only justify frisking (nervousness, anger, avoiding eye contact, etc.), the car can not be searched or inventoried (unless illegal items are found during the frisk).
As of writing this, there are no indicators and no methods to detect suspicious items (as of writing this) inside of a vehicle in plain view during a traffic stop or accidents.
Currently, there is no way to ask for consent to a vehicular search or a person search.
Right now, there is no way for someone else to pick up an arrested suspect's car. So, the best option to deal with that car is to tow it (and therefore inventory it before towing).