There is at least one LSA that is approved for IFR flight: The Zodiac 650 (available as a kit, or a factory-built Special Light Sport Aircraft).
More broadly, you can operate light sport aircraft under IFR (in VMC or IMC) as long as it's permitted by the operating instructions. AOPA's Light Sport / Sport Pilot FAQ addresses this well, so I'll just steal their answer:
Can I fly a special light sport aircraft (SLSA) in IFR conditions or at night?
Only day/VFR conditions are specifically addressed in the ASTM consensus standards that govern the production of SLSA. Being that sport pilots and those exercising sport pilot privileges are limited to flying only in day/VFR conditions, this seems appropriate.
On the other hand, if an appropriately rated pilot (example: private pilot with an instrument rating) wants to fly SLSA under IFR or at night, the aircraft's operating limitations must allow it, and the aircraft must be equipped per 91.205 for VFR flight at night and/or IFR flight. Additionally, 91.327(d) requires all SLSA to be operated in accordance with the aircraft's operating instructions. Operating instructions differ from operating limitations in that the engine, airframe, and accessory manufacturers issue them; the FAA issues operating limitations.
An example of operating instructions is a SLSA equipped with a Rotax engine. Rotax's operating instructions prohibit the use of a Rotax engine at night or in IFR conditions unless it is the FAA type certificated engine (14 CFR part 33). Other engine, airframe, and accessory manufacturers might impose similar restrictions.
If you are appropriately rated and would like to operate a special light sport aircraft at night or under IFR, contact the manufacturer to determine if any provisions can be made.
(Note also that there were some rumors of the ASTM consensus standards changing at some point to require
DAY VFR ONLY placards on new Special Light-Sport Aircraft -- At this time I don't believe that's happened yet, but it might...)