Yes, in principle, but while we have indeed found some extremely low-metallicity stars, we have yet to discover zero-metallicity stars. Moreover, accretion of metal-rich gas over billions of years may actually increase a star's metallicity significantly.
The initial mass function
Stars are born from clouds that fragment into clumps of a distribution of sizes, and so the initial mass of a population of stars follow a distribution of masses known as the initial mass function (IMF). In the Milky Way, at least in our neighborhood, a good approximation is given by the Salpeter (1955) IMF, but stellar populations in other regions and galaxies have been fitted better by other IMFs (e.g. Scalo 1986; Kroupa 2001; Chabrier 2003).
In general, the IMF is a power law-ish function with a peak around $\lesssim 1 M_\odot$, and with a lower threshold mass of $\sim 0.1\,M_\odot$ (needed for nuclear fusion). In principle there's no upper limit, but the function decreases so steeply that stars with high masses quickly become very rare. Such IMFs are called bottom-heavy and top-light, because it is "heavy" (has a high value) in its "bottom" (i.e. at low masses), and it is "light" in its "top".
Population III stars
There are reasons to believe that, in the early Universe, stellar populations are described by more top-heavy IMFs (e.g. Steinhardt et al. (2022)). Indeed, the very first populations are hypothesized to be characterized by the extremely massive stars that you refer to, known as Pop III stars. But even very top-heavy IMFs are still a distribution — albeit much more flat (logarithmically) — and hence there could also be low-mass Pop III stars.
The reason that Pop III stars are so massive is that they are metal-free and hence lack the effective cooling given by the many electronic transitions of metals. To collapse to a star, the mass $M$ of a gas cloud must exceed a certain threshold known as the Jeans mass which is proportional to $T^{3/2} / \rho^{1/2}$, where $T$ is the temperature and $\rho$ is the density. Thus, if $T$ stays high, a cloud of a given (over)density must have a high mass.
Numerical simulations
Insight into Pop III IMFs has come from analytical calculations, but mostly from numerical simulations because you need to take into account many things that are only analytical in idealized cases, e.g. turbulence, radiation field, clumpiness, star formation efficiency, formation of binaries, magnetic fields, and more. In the early Universe, the first clumps to collapse were of the order $10^{5\text{–}6}\,M_\odot$, but to simulate the formation of such a clump in a cosmological context, and at the same time have the resolution to resolve single stars is a challenging task which has yet to be done satisfactorily.
The figure below compares typical IMFs (left; from Ivan Baldry's website) to a variety of predicted Pop III IMFs (right; from the recent review by Klessen & Glover 2023):
The take-away from the right plot is that, 1) as noted above, some IMFs extend to rather low masses, and 2) as noted in the Klessen & Glover paper, "our understanding of the IMF of primordial stars is still quite limited". Although some of the IMFs do go down to very low masses, the ones that included radiative feedback (which heats the surrounding gas and hence also impede star formation) seem to truncate at $M \sim 10\,M_\odot$.
Low-mass Pop III stars
As the first supernovae go off in an early clump of gas (and dark matter), they may impede further star formation in that clump, but even if they don't, they quickly pollute the gas with metals. This means that, yes, formation of Pop III is only possible during the dawn of the Universe. But given our ignorance of the Pop III IMF, in principle it is possible to form low-mass stars, and thus you're right that
If a red dwarf could form early on, they'd still be around today, since their lifetimes are in the trillions of years.
If gas accretion is slow, this seems to help low-mass star formation. Thus, from high-resolution simulations Stacy & Bromm (2014) predict "as a very rough estimate that one out of a few tens to one out of 100 minihalos will host a low-mass system", and that "it is conceivable that on the order of thousands of Pop III stars from such low-mass systems may exist in the nearby Galactic halo".
So why haven't we found them? I don't know so much about this, but part from the fact that, if we are to believe Stacy & Bromm, only one in 100 million of the Milky Way's stars should be a Pop III star, the problem is that red dwarfs are not very luminous, and so not easy to see too far away. But while we haven't found any "real" Pop III stars, we have found some that come pretty close, e.g
SDSS J102915+172927 with a metallicity of $Z/Z_\odot = 10^{-6.16}$ (Caffau et al. 2011),
SMSS J160540.18-144323.1 with $Z/Z_\odot = 10^{-6.2\pm0.2}$ (Nordlander et al. 2019), and
J0023+0307 with $Z/Z_\odot < 10^{-6.6}$ (Aguado et al. 2018) (in all three examples, the metallicity is defined as the abundance of iron, and are then normalized to the Solar abundance).
Accretion over time
@ProfRob raises a good point in the comment section: Even if a star is born in the early Universe with zero metals, as it travels through the interstellar medium (ISM) of its galaxy over the billions of years to follow it will slowly accrete matter and hence metals.
Could that affect its metallicity? Let's do a rough estimate:
To first order, the appropriate accretion rate can be calculated as the so-called Bondi-Hoyle-Lyttleton accretion (Hoyle & Lyttleton 1942; Bondi 1952).
As a mass $M$ travels through a gas of density $\rho$, its gravity will form a "wake" of gas behind it, some of which may accrete. Thus, the star not only sweeps up gas according to its own cross section, but a much larger area given by the distance from the star where the escape velocity equals the speed of sound $c_\mathrm{s}$. The resulting accretion rate is then roughly
$$
\frac{dM}{dt} = \frac{\pi\rho G^2 M^2}{(v^2 + c_\mathrm{s}^2)^{3/2}},
$$
where $G$ is the gravitational constant, and $v$ is the speed of the star through the ISM.
Typical densities in the ISM are $n\sim 1\,\mathrm{cm}^{-3}$, while both stellar velocities and the speed of sound are of the order $10\,\mathrm{km}\,\mathrm{s}^{-1}$.
In this case, the equation evaluates to $dM/dt \sim 10^{-8}\,M_\odot\,\mathrm{Gyr}^{-1}$ for a $0.1\,M_\odot$ star.
With an average ISM metallicity of 0.01 (0 when the star is born, 0.02 in the present-day Universe), during 10 Gyr the star will accrete $10^{-9}\,M_\odot$ of metals and hence increase its metallicity to $Z \sim 10^{-8}$, or a few times $Z/Z_\odot \sim 10^{-7}$, which would bring it near observed low-metallicity stars.
However, Pop III stars are unlikely to take part of the disk's ordered motion, instead forming a halo around the disk. Not only does this mean that they spend most of its time in lower-density environments, but it also means that their velocities are an order of magnitude higher.
This means that accretion is most likely negligible.
