The Republicans believed the Alien and Sedition Acts violated the Bill of Rights; name and describe another issue from this era that may have also challenged the new constitution.

The bitter disagreements between the Democratic-Republicans and the Federalist Party came to a head with the passing of the Alien and Sedition Acts: a series of successive legislative acts passed by a majority Federalist congress and ratified by Federalist president John Adams in 1798. The Acts gave the government broad powers to deport or jail recent immigrants and to jail or fine people for making false statements about the government. The implication of this legislation was not insignificant to the Democratic-Republicans, as their base was primarily composed of immigrants.
The laws pertaining to "sedition" (or defiance of authority) therefore resulted in the arrest and prosecution of several prominent newspaper publishers and politicians. This was in open defiance of the Bill of Rights (passed less than a decade before), which guaranteed freedom of speech. However, the Alien and Sedition Acts were passed before the practice of judicial review had been established, which meant that the Supreme Court did not yet weigh in on the constitutionality of laws passed by Congress. In the absence of this check, a standoff resulted between the two parties.
The Alien and Sedition Acts weren't the only challenge to the constitutional protections at the time, and many other issues tested the limits of the protections afforded by the fledgling Bill of Rights. Most notably, the issue of slavery presented an immediate contradiction to the new constitution and Bill of Rights, but this was an issue that would be largely avoided for the next half-century. In fact, the constitution explicitly protected slavery (i.e. the "Three Fifths Compromise"), and lawmakers avoided the issue of defining which people were to be included in the protections afforded by the Bill of Rights.
Soon after independence, and even before the formation of the US constitution, many states began forming their own constitutions; in some northern states, the abolition of slavery followed as a result of these constitutions. For instance, in Massachusetts, a woman by the name of Elizabeth Freeman (or "Mum Betts")--having overheard her owner and other lawyers discussing the liberty and freedoms that all men are born with--successfully argued for her own freedom in the early 1780s. Her case, along with cases like those of Quock Walker, established that slavery was incompatible with the Massachusetts constitution. A key figure in these cases was Justice William Cushing, the Massachusetts Supreme Judicial Court Chief, who would later go on to be appointed to the Supreme Court of the United States by George Washington.
As seen in these examples, a clear conflict was illustrated between the inalienable rights set forth in the Bill of Rights and the practice of slavery. This conflict would continue to simmer until the eruption of the Civil War, and subsequently the passage of the Thirteenth Amendment--which asserted that, at least on paper, slavery was unconstitutional.