Yes, it is possible to seek change from L to H while still in the U.S. However, the start date of the H-1B would be October 1st of this year. As long as your L-1B is set to expire later than that date, you should be fine. Alternatively, you could seek consular processing, but would have to leave the country prior to the October 1st start date and reenter with an H-1B visa.