Can I rent a car at the airport without a flight in Lisbon?

Yes. I travel through Lisbon airport regularly. The car rental desks are in the terminal and you can easily just show up and hire a car. It is typically more expensive than renting elsewhere in Lisbon (as mentioned by others). Expect to queue, it is often very busy and all the companies seem to be very slow.

You can rent a car at the airport despite of where are you coming from. It doesn’t matter. What Im not sure is if the rates are the same in the city centre and the airport. What you can do is call the agencies at the airport and ask for prices.
There are car rental desks in the Arrivals area of Terminal 1. Open from 6:00 am to 1:00 am.

Avis/Budget (Tel +351 218 435 550)
Europcar (Tel +351 218 401 176)
Goldcar/Rhodium (Tel +351 218 413 768)
Guerin/Enterprise (Tel +351 218 486 191)
Hertz (Tel +351 219 426 300)
Sixt (Tel +351 255 788 199)

Source: https://www.airportsdata.net/europe/portugal/airports/lisbon.php

I, too, have done this on several occasions, although not in Lisbon.

Note: Sometimes it is less expensive to rent from a downtown office of the same company because of airport fees. On the other hand, in New York City, I found the best deal by far available only at the airport office. You should definitely use a travel site to check this unless you are indifferent to the rate charged.

You can rent in the airport in any situation. It doesn’t matter if you’re flying in, out or not even flying at all.

I know this for a fact. I have some family that has rented in Lisbon, in a couple of different offices/brands, and they weren’t neither flying in or out. Rules tend to be similar accross rental offices.

The business of these companies is rental. The airport is just a convenient place to be since there is demand. They don’t really care where their clients come from. What might happen is that often rental companies at the airport ask for a flight number on their online reservation forms. Don’t be discouraged by that. It’s only relevant for them to know if your flight is delayed and deal with that fact (if you’re actually flying).
Dealing with the fact might mean several things:

• If your flight has arrived and you don’t show up after a while they might call to know if you still want the car.
• Or if they pick you up at the airport, they will only meet you upon flight arrival. With the flight number they know the time of arrival and if there are delays.

If such field exists either ignore it, write not applicable or n/a (short form).
If they are strict about it just fill in something (eg.: aa1234) and drop an email telling you’re not arriving by plane since you’re already in the city. Usually they don’t use this field as validation for the reservation. They most likely already charged your credit card and that is by far more relevant. As a general rule it’s still not a bad idea to drop a message. Specially if there is a shuttle that might wait for you.

It doesn’t matter.

I have rented cars at airports without flying many times, including in Europe, such at Belfast City Airport in Northern Ireland (United Kingdom), and Luleå Airport in Sweden. I’ve found that those may or may not be cheaper, but almost always have longer opening hours and a wider offer, compared to city car rental places — if the latter even exists locally.

In Belfast, I arrived for pick-up before the first flight of the day had even arrived. In Luleå, I arrived at a time of day that no flight had arrived for a couple of hours. In both cases, there was no queue and the arrival terminal was deserted. In both cases, there were absolutely no questions asked, and the idea that there might be an issue frankly never crossed my mind.

It doesn’t matter.

I can’t speak for Lisbon, but i have done exactly the same in the US successfully. Nobody asked for any flight details, when i picked up or dropped off the car. The only condition some rental car companies have, is when you only have a debit card. Then they might require to provide actual flight confirmations. If you use a credit card for deposit, it’s not required (i know that Hertz has this condition, when renting in the US).

However, you might want to compare the fares with city stations, since e.g. in Germany the stations at airports or train stations always charge a significant fee, so that might be the same in Lisbon.