Any airplane can take off without flaps … eventually.
The main reason for taking off with flaps partially or full extended is to reduce the amount of time and distance required to leave the runway. The flip side of this is that flaps increase drag a lot and make it more difficult to rapidly build airspeed after leaving the ground.
Large airliners usually take off with flaps partially extended, and in fact any high-speed aircraft may use flaps at takeoff, because airplane designs that fly well at high speed often have poor performance at low speeds. Low-speed aircraft don't usually need flaps to take off because they have good low-speed performance (since that's the only speed they have!).
If your runway is long enough, you can take off without flaps in any aircraft. Most airports don't have runways long enough for that when it comes to jets, so the jets have to set some degree of flaps before taking off.
There are sometimes other reasons for taking off with flaps. One example is speed restrictions for the tires. You might be able to take off without flaps, but only at a speed that is higher than the tires are certified for. This effectively forces you to use flaps at takeoff except for emergencies.