Regular imperfective counterparts for стать, сесть, лечь should have been *ставать, *седать, *легать, but those forms have fallen out of use as standalone words in the codified language, though they continue to exist as roots: встать / вставать, присесть / приседать, налечь / налегать.
These forms have been suppleted with the reflexive causatives: -становить, -садить, -ложить are causatives meaning "make stand", "make sit" and "make lie", respectively, and when used with a reflexive postfix they mean "make oneself stand", "make oneself sit" and "make oneself lie".
I can't think of any more suppletive imperfective verbs with this exact paradigm (causative + reflexive) in Russian, but there are other paradigms: vowel alternation (умирать / умереть, собирать / собрать); stress alternation (насыпа́ть / насы́пать, нареза́ть / наре́зать); good old replacement with etymologically unrelated roots (ловить / поймать, брать / взять) etc.