Is it feasible that a sitting president's approval rating be so low that his/her own party would choose to nominate someone else to run for the office? Has this ever happened?
It's fairly common throughout US history actually. 9* out of the 58 US presidential elections have turned out this way.
9th president William Henry Harrison won the 1840 election on the back of the Whig Party, but the party was young—only 5 years old—and fragile, being primarily united by its opposition to the Democrats rather than any consistent ideology of its own. When Harrison died just a month after his inauguration, he was succeeded by John Tyler, who almost immediately came into conflict with his own party. This culminated in Tyler being kicked out of the Whig Party in 1841, and a failed attempt by the Whigs to have him impeached the following year. Hated by most of his own party, he launched an independent reelection bid, but dropped out of the race in August of 1844. Most of his supporters voted for the Democrats, resulting in Democratic nominee James K. Polk narrowly winning the election.
Polk had pledged on the campaign trail to serve only a single term. So he did just that. Lewis Cass won the party nomination in his place, but lost to Whig Party candidate Zachary Taylor.
13th president Millard Fillmore, a Whig, was extremely controversial even within his own party on the eve of the 1852 election. This was reflective of the growing divisions within the Whig Party over the issue of slavery. Whigs in northern states, who supported a strong anti-slavery stance, were furious at Fillmore’s Compromise of 1850, which allowed California to be admitted as a free state while granting several concessions to slave states. Fillmore’s moderate stance was unpopular, and anti-slavery Winfield Scott received the Whig nomination. He was defeated to pro-slavery Democratic candidate Franklin Pierce.
The slavery issue went from being one of many major political issues to the single focal political issue in the years that followed. The Whig Party entirely collapsed, with the radically anti-slavery Republican Party and nationalist American Party appearing in its wake. In the Democratic Party, 14th president Franklin Pierce had not been a popular president, and lost the nomination to James Buchanan. Buchanan would go on to win the 1856 election in a landslide despite only receiving 45% of the vote, in large part due to fracturing of the Whig Party.
As the slavery issue became even more divisive, serious talk of Southern states seceding from the United States emerged in the 1850s. This created the conditions of the election of 1860. The Democratic Party split into the ‘Northern Democrats’ and ‘Southern Democrats’, the former favoring a moderate stance on slavery and the later being more explicitly pro-slavery. The Constitutional Union Party developed from the earlier American Party, trying to downplay the issue of slavery for the sake of political stability. In any case, James Buchanan did not run for reelection, being incredibly unpopular. Ultimately, Republican candidate Abraham Lincoln became president, one of the final steps on Americas path to Civil War.
17th president Andrew Johnson, of the Republican Party, was unpopular to the point of being impeached, battling even with his own party. Unsurprisingly, he lost his party’s nomination to war hero Ulysses Grant, who went on to win the election of 1868.
19th president, Republican Rutherford B. Hayes, had pledged to only serve for one term. He remained true to his promise, and James A. Garfield won the 1880 election in his place. Similarly, 30th president Calvin Coolidge chose not to run for reelection in 1928, with fellow Republican Herbert Hoover winning the election in a landslide.
The most recent time that an incumbent president failed to receive his party nomination was in 1968, when 36th president, Democrat Lyndon B. Johnson, chose not to run for reelection in light of his horrible 35% approval rating in January of that year—largely a result of his handling of the Vietnam War.