I'd have to agree with drakensis. See the link for current orbital insertion equipment.
https://www.sarna.net/wiki/Drop_PodOther than that, there is two existing ways I can think of.
One, the original Kyushu fluff from TRO3057 stated in the fluff (page 124 final paragraph, see below) that specially modified small craft could individually transit Mechs from warship to ground (it does not say they can reverse the process) but I've never seen this application put into practice. I believe the same fluff also indicated that this process was a long one and hence not efficient in a combat setting.
The Kyushu’s 108 BattleMech cubicles are perhaps its most unusual feature. These cubicles are located in the vessel’s large cargo section and can accommodate a full regiment of war machines. The Kyushu’s design enables its ’Mech cargo to descend to a planet's surface in one of three ways. The first and most common is via DropShips, which can dock at one of the four hardpoints situated along the Kyushu’s spine. The second option is one of the Kyushu’s twelve specially modified cargo shuttles. These craft are lightly armed and armored and can transport two ’Mechs at a time. Finally, if the frigate is in low orbit, ’Mechs can be dropped directly from the Kyushu's bay doors.
Two, the widespread deployment of a LAM force (i.e. the Marik Guard) which could theoretically enter and exit a planetary atmosphere without the need of 'dropships' is possible but to my knowledge never before tactically applied at a regimental level. The Marik Guard was by the time of the 4th Succession War so over protective of its rare equipment that it dare not even attempt rapid insertion and departure. The SLDF could have utilized battalion sized contingents in this fashion but again I don't believe this was ever detailed. Not to say it hasn't happened but I've never heard of it. Long ago in my home campaigns a large created baddy faction named the Imperium did indeed do this but in a stand up fight versus BattleMechs the LAMs (which had an 80 ton max) were poorly matched as their conversion equipment and faster nature (large engines) occupies enough design space to make them lighter armed. However, as a rapid deployment force using movement they made superior raiders and terrifying harassers.
My undeveloped 'Mech Marine' concept that was originally destined for Shattered Dawn would also utilize LAM forces and rapid insertion/extraction methods to specialize certain forces in this fashion. Securing initial planetary beach heads and engaging in paradrop operations would be their calling cards in endoatmospheric operations as well as engaging in aquatic combat from their origins. Exoatmospheric (zero-G operations, warship hull defense, etc.) and exoticatmospheric (other planetary environments not Terran standard) operations would also be within their scope. These three AtmoEs would make them your friendly non-standard Mech operating warfare practitioners of House 'X'. Far beyond your militia force that has never seen any terrain outside its homeworld and even your standard Mech regiment that does journey from system to system operating.
Finally, another quick retrieval system I've thought about is a 'skyhook' for Mechs but in practice not sure how that could work.
https://en.wikipedia.org/wiki/Fulton_surface-to-air_recovery_system
