In the election of 1824 Jackson won a clear plurality of the popular vote and in the electoral college. Yet, because Jackson did not win an outright majority, the House of Representatives decided the election's outcome. Led by Speaker of the House, Henry Clay, the congressmen chose John Q. Adams instead of Jackson. When Adams named Clay his Secretary of State, Jackson and his allies labelled the election a "corrupt bargain."
At that point did Jackson have a feasible opportunity to seize power? Did he restrain himself from doing so?
A few points to consider:
Jackson won a plurality of the popular vote and in the electoral college.
The election's official outcome in favor of Adams was viewed by many as a "corrupt bargain."
Jackson was a national war hero renowned for his military prowess and people still called him General Jackson. These points suggest that Jackson had support from ordinary citizens and the military while his opponents were in disarray and possibly colluded to steal the election of 1824.
I realize Jackson's whole thing was preserving the Federal Union but did he or others (supporters and opponents) consider toppling the republic?
I study foreign policy, not internal squabbles, but having read Daniel Walker Howe's great book What God Hath Wrought, along with numerous other texts that refer to this election, I never once got the impression that revolt was even on the table. Jacksonians were extremely displeased, no doubt about it, but I don't think they considered this an option.
A military coup wasn't really practical, in addition to being extremely destabilizing and potentially alienating for Jackson's supporters. A civil war was something no one wanted, especially after the scare that led to the Missouri Compromise in 1820. An octogenarian Thomas Jefferson had said that the compromise was "like a fire bell in the night, awakened and filled me with terror". People didn't really want war and as popular as Jackson was, there doesn't really seem to have been a propitious climate for outright rebellion.