Doubtful. And that isn't a knock against Rocket Lab, targeting a different niche is smart business, but still doubtful. Because the target of any rocket under development right now that wants to directly compete with SpaceX can't be F9, it needs to be Starship. The fundamentals along with the iterative capability and in-house demand built into the core of the design plan indicate they're going to be about to push costs below $200/kg and eventually even below $100/kg, along with enormous other sets of capabilities. That will represent just a mind blowing paradigm shift in cost to LEO, and SpaceX will probably want to retire F9 completely as soon as they're done with contracts and SS/SH is fully certified.
Rocket Lab should still be able to find a healthy market, like now, for customers who want specific orbits and times for smaller payloads that don't line up well with ride sharing on a big rocket. But it just won't be in the same market slice at all. Which is fine! The entire space market is set to grow a lot, no zero-sum games for a good long while, and in a growing market there can be room for many players to grow together. But everyone not on the leading edge is going to have to stay nimble.