SpaceX is privately held company so currently there are no way to buy it's shares on public stock market. But it's not a reason to give up. There are some companies (SpaceX clients) which can also make profits from SpaceX's successes.
Here some of them:
ECHOSTAR CORPORATION
NASA- MULTIPLE SUPPLY MISSIONS
IRIDIUM
SKY PERFECT JSAT CORPORATION (JAPAN)
EUTELSAT AND ABS
THAICOM 8
SES
NASA (JASON-3)
ORBCOMM
THALES ALENIA SPACE
ASIA BROADCAST SATELLITE/EUTELSAT
U.S. AIR FORCE (DSCOVR)
ASIASAT
ORBCOMM
THAICOM (THAILAND)
MDA CORP. (CANADA)
FATSB (MALAYSIA)
KWAJALEIN
U.S. GOVERNMENT, ATSB AND NASA
AIRBUS DEFENCE AND SPACE
ARABSAT (ARABSAT 6A)
BANGABANDHU-1
BIGELOW AEROSPACE
BULGARIA SAT
CONAE (ARGENTINA)
ES'HAILSAT
GLOBAL IP
HISDESAT
INMARSAT
IRIDIUM
KOREASAT
NASA RESUPPLY MISSIONS- MULTIPLE
NASA CREW MISSIONS- MULTIPLE
NORTHROP GRUMMAN
NSPO (TAIWAN)
OHB SYSTEM AG
RADARSAT
SES
SPACEFLIGHT SERVICES
TELESAT
U.S. AIR FORCE
VIASAT
TODO: Add tickers.