Check to be sure that contacts are turned on in Settings>iCloud on your phone. If you are syncing contacts with multiple accounts on your phone, also go to Settings>Mail,Contacts,Calendars>Default Account (in the contacts section) and make sure it's set to iCloud. If you made any changes, test your contacts again to see if they sync now.
If all that's correct and you still can't access the contacts from icloud.com, go to Settings>iCloud on your phone, turn Contacts off, choose Delete from My iPhone when prompted (your iCloud contacts will still be in iCloud) then turn contacts back on again and see if they download down to your phone now. If they don't, force-close the contacts app (double-tap the home button, locate and swipe up on the contacts app to close it, tap the home button again), then reset your phone (hold the sleep/wake and home buttons at the same time until you see the Apple logo, then release). After it restarts, test your contacts again.
If it still isn't syncing, save any my photo stream photos that you want to keep on your phone to your camera roll (unless already there) by opening your my photo stream album, tapping Select, tapping the photos, tap the share icon (box with upward facing arrow), then tapping Save Image (or Save to Camera Roll if you aren’t running iOS 8), then go to Settings>iCloud, tap Sign Out (or Delete Account), choose Delete from My iPhone when prompted (your data will still be in iCloud), then sign back in (your data will download back to your phone). Then test your contacts again.