Saying "would have" is too definite, but there was at least some support for the idea of forcibly taking much of the Louisiana territory.
In 1795, the United States and Spain, which controlled Louisiana at that time, signed Pinckney's Treaty. This treaty, amongst other things, not only defined the borders between the USA and the Spanish territories, but also granted right-of-way to US ships navigating the Mississippi River and a right of deposit in New Orleans. This meant that American merchants could freely ship goods on the Mississippi and transfer them through New Orleans without paying export duties.
France took control of the territory back in 1802, and at about the same time, the acting intendant of Louisana revoked that right of deposit. I don't believe it's entirely clear why he did so, but he believed at the time he was acting on Napoleon's orders. Yet, when selling the territory to the United States, part of Napoleon's reasoning was that the revocation of the right of deposit had made peaceful occupation impossible.
At any rate, that revocation incensed American opinion. Alexander Hamilton wrote that the United States should "seize at once on the Floridas and New Orleans and then negotiate." The two American consuls in New Orleans, Daniel Clark Jr. and William E. Hulings, wrote that the United States could take the territory without any bloodshed. And in Congress, Senator James Ross of Pennsylvania (a Federalist) introduced a resolution authorizing the deployment of 50,000 militiamen to take New Orleans. The Democratic-Republicans also supported a resolution that would've put 80,000 armed men on standby.
However, Clark's, Hulling's, and Hamilton's views were decidedly in the minority as all resolutions to take the nation to war were soundly defeated in Congress. So though people were upset, it's not clear that the nation was willing to go to war over the matter of free navigation of the Mississippi. Moreover, the right of deposit was restored in May 1803 -- just seven months after it was revoked, although Americans now had to pay an export duty.
Through all this, Jefferson preached patience and reasoned that with war in Europe, France would be open to a purchase offer. He thus sent Monroe to negotiate such an offer, and Napoleon not only accepted, but allowed the United States to buy all of France's holdings in North America. Worth noting though that Jefferson also gave Monroe instructions that in case he failed, he should then seek an alliance with Britain to keep France out of the Southwest (at that time, Louisiana and West Florida).
Source:
My primary source for all this is "The Mississippi Territory and the Southwest Frontier, 1795-1817" By Robert Haynes, pages 65-69.
This question is what if history territory, but in my view it's worth considering 4 factors:
the ideology of manifest destiny
the monroe doctrine
demographics trends
earlier and more extensive industrialization in America
These provide a framework to the view the question with, as they offer causes for conflict and hints at the possible outcome.
I don't think factor 1 needs much explaining, other than that it's unlikely the tentative alliance between America and France would have shielded France from american aspirations long term. The french revolution and especially so called Terror was prosecuted in such a way that it disgusted many of the intelligentsia and political leaders in the United States and relations were cool enough to have cut off trade to both France and Britain in order to stay neutral and to actively lobby for safe passage to America for enemies of the then french government like Lafayette. Depending on when we are talking about, the stance of the US towards France could even be quite hostile over the large French claimed territories that border it, especially if it is under a French government that is seen as reactionary and/or despotic and a threat to the revolutionary United States, like say Emperor Napoleon III. This stance might even be partially predicated on settlers going into french territory to claim the land themselves without initial federal directives, perhaps with the collusion of state governments, as happened with the Cherokee territory with Georgia and georgian farmers.
This brings us to factor 2, which is the monroe doctrine, which was a stated policy and geopolitical aim of the early United States to keep european colonial powers from returning to power in the new world in the newly independent states in the western hemisphere and to safe guard the independence of such states through force. This policy goal came into force in the american intervention into the Cuban war for secession from Spain. The United States was not going to tolerate european colonial powers from being in striking range of the United States, and having a land border with France would have been a high priority target. Especially in a scenario like the French invasion of Mexico in order to over through the Mexican Republic and install a monarch by force. This would have been tantamount to a declaration of war on the United States as well, and in real life the reason why Lincoln didn't intervene in this affair was because the United States was busy with its civil war. In a timeline without the Louisiana Purchase, do you get tensions over slavery being introduced into the new states the same way? Does it develop into outright civil war? Could the US intervene in the French invasion of Mexico and remove the French from blocking their way to the west? No matter what though, the US would almost certainly seek war with France over these territories because they pose an existential threat, as a staging area.
Factor 3 is a really important one, because as they say, demographics are destiny. America was the recipient of numerous waves of immigration of people either hoping to have religious freedom, political freedom, the prospects of land ownership, or possible wealth that might be unattainable in their previous residence. In no conceivable way will this factor be less relevant, and it's in addition to a generally high birth rate anyway. Concurrently in France, you have a country with enormous population loss from the almost unending wars from 1792-1815, including its own massacres on its own population like with the Vendeé uprising. This lost generation is coupled with the relatively lower birth rate of France not only in comparison to America, but to pretty much all its neighbors. The longer the wait before such a war between the two powers, the weaker the position of France is, especially since it's highly unlikely their territories on mainland North America would ever be densely settled in order to support a heavy military presence.
Factor 4 also points towards a hypothetical american victory depending on when we are talking about. The US because of the economic prospects it offered as a yet unbuilt land was also attractive for engineers like Samuel Slater who brought early industrial textile production techniques to the US, but there were also "indigenous" innovations as well such as the cotton gin, the steam ship, Evans' automated flour miller, a domestic system of interchangeable machine parts, etc. These kinds of economic developments would come slower to France. In addition, the US also went on to develop a highly developed rail network, which of course is a huge boon for military conflicts, and would make the seizure of French Louisiana all the easier.
This is all also glossing over what other powers responses might be towards France's territories in North America, especially during the Napoleonic wars. It's very possible that French Louisiana would eventually be a theater of war in this scenario with the British hoping to seize the colonies there, using British Canada as their staging point. The British would probably succeed and would probably demand at least a substantial portion of them in the Congress of Vienna. But this question concerns hypothetical US seizure of French Louisiana from France so I'm not touching the full extent of how everything else could go in this timeline, because that kind of exercise is endless in scope.
I want to ask a counterfactual follow-up, but I'll try a more legitimate answerable question first:
To what extent did President Adams or the Federalists have designs on capturing Louisiana/New Orleans? What are the chances they would have attempted to take it had they won the election of 1800?
Always be skeptical of things like this is history. From modern day it's easy for anyone to take their worldview and cherrypick the infinite set of data points we call history to fit it. People of the future always talk about history in terms of who can get the closest idea to what it was like being there to make their opinion of something. But of anyone, I'd say the people living in the US at the time knew the least about what was going to happen if France didn't sell.