The 1841 British edition accidentally replaced the modal may with might. The subsequent LDS edition (1849) restored the correct may. For another example of variation between may and might, see the nearby discussion under Alma 22:7. For a general list of cases of this variation, see under Jacob 5:13. For each case of may /might, the critical text will follow the earliest extant reading.
Summary: Maintain the modal verb may in Alma 22:15, the earliest reading (“that I may receive this great joy”).