Palin talks about "breaking up a monopoly" of the big oil companies in AK. Not clear what she means by "monopoly," but she has pushed to get new companies (besides the big three already in AK) involved in the construction of a proposed new gas pipeline. That pipeline has not yet been built, and it's not clear it can get built without the expertise of the big three.
In broader terms, this is her record of "opposing" the oil companies:
Palin's predecessor, Governor Frank Murkowski, had negotiated a change to the state's production tax on oil. It was negotiated with the big three oil companies, then presented to the state legislature. (It wasn't a simple increase or decrease — it was a change in how the tax was calculated, and the final result would depend on market prices, etc.) The legislature passed it (after contentious debate).
Later that year (fall 2006), the FBI raided several Republican legislators' offices, the first sign of what would become the VECO oil company bribery scandal. Palin was elected a couple months later. She re-opened the issue of the oil production tax, and it was raised slightly (though I think — think — they stuck with the same basic method of calculating it).
Really, the FBI did the heavy lifting, politically.