Venture-Lieutenant, Finland—Turku
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Probably not, because PFS states:
A cavalier can pledge to a legal deity, religion, or pantheon, an organization they have access to, or the Pathfinder Society.
and druid orders aren't a legal deity, nor a religion, nor a pantheon, and probably don't count as an "organization" - otherwise, a druid of (any random order) could purchase secondary initiation to gain access to the feats of another order, and that seems unintended.
Additionally:
Druids of the specifically listed orders may also take the other elementals (e.g. a wave order druid may take a water elemental companion).
You need to be a druid to gain access to them (emphasis is mine), just pledging for the order isn't enough.
Further, the elemental companions don't have the mount trait as required by the cavalier dedication (although it's a bit ambigious whether " animal companions with the mount special ability, as well as any additional options from your pledge" means that you can choose OTHER THAN ANIMAL companions That Still Have the Mount Special Ability, or if it means you can choose other companions regardless of whether they have the mount ability.).
Even if you could, the elementals would still be small and would get upgraded to medium, and since the companion needs to be one size bigger than you, this would only work for a small or tiny cavalier.
There are so many different points of conflict that it doesn't seem intended. A small (or tiny) druid cavalier that belongs to (and has pledged to) an elemental order might be able to take an elemental companion, but it definitely doesn't look like a default option for any and all cavaliers who want to ride a flaming rat.