Three problems. First, Andrew doesn't know exactly where it is, and even if he sent an expedition to look for a teapot-shaped rock formation in the west, the oil field is actually several miles from Teapot Rock.
Second, Wyoming wasn't even a territory at the time; transportation would be a severe handicap. Of course he could build a railroad eventually, but that would take years.
Third, from what I could find, Teapot Dome has always had a low production rate, and a lot of the oil was not economically recoverable until recently.
Also, he's going to have to convince the US to build an actual navy. We'll need it to challenge the British for colonial dominance in the Middle East and Asia. Otherwise, we're dependent on foreign powers for oil, rubber, and everything else.
Andrew is in favor of building a large, modern navy, but not for reasons of empire.
Even with greatly accelerated industrialization, and earlier conversion from coal to oil, the US would still have enough petroleum reserves to meet domestic demand well into the 20th Century (a point to which
Delvo alluded above).
As for rubber, Andrew will buy up and greatly expand most of the rubber plantations in Brazil, and ensure that they are operated humanely, rather than using the indigenous population as slave labor. This will not be as economically efficient as British colonial rubber production, but it will be adequate to meet demand until synthetics can be developed.
Actually, they're probably better off sailing to England and starting their world domination there.
Andrew and Jenny have no interest in world domination. Some of their major goals, not necessarily in order of priority (though the first two are):
- ensuring that the Union wins the Civil War
- ending slavery
- reducing casualties and property damage from the Civil War
- improving public health and reducing infant mortality
- promoting economic and political equality for women and minorities
- promoting religious and ethnic tolerance and understanding
- promoting freedom, democracy, and eventual decolonization
- promoting world peace
- retarding the spread of communism, authoritarianism, monarchism, imperialism, and other anti-democratic ideologies
- fighting corruption, both in and out of government
- improving public education
- reducing poverty
Note that while Andrew and Jenny agree on all these goals, their priorities are not identical. For example, Jenny is more interested in promoting health and women's rights, for obvious reasons.
Further, even if they were interested in world domination, taking the long view, I believe the US would provide a better platform for that over the period 1865-1895.