As far as I know gas giants have no metals. How did the Dwellers get into space in the first place? Are they capable of being in the vacuum of space without ships? I don't remember Banks talking about that. It seems like a bit if a chicken and egg problem as to how they could have gotten metals and minerals to create their ships in the first place.
The original Dwellers bootstrapped their civilization using meteoritic iron and other metals:
A classic water-world environment had almost no easily available metals. Any metal-bearing ores that a waterworld possessed tended to be locked away under all that ice, deep in the planet's inaccessible rocky core. Waterworlders had to do what they could with what fell from the sky in the shape of meteorites, and in this shared a developmental background with gas-giant Dwellers.
Backing this up, when Fassin is burying his companion/superior Colonel Hatherence it notes that iron, at least historically, was rare:
Iron had always been and, sentimentally, ceremonially, still was a semi-precious metal for Dwellers.
Of course as a technological species they can get metals from other planetary bodies and, potentially, deep within their worlds. (Nasqueron does have a rocky core, and presumably metals in it.)
There was a solid rocky core to the planet, a spherical mass of about ten Earth-sized planets buried under seventy thousand vertical kilometres of hydrogen, helium and ice, and there were purists who would call the transition region between that stony kernel and the high-temperature, high-pressure water ice above it the planet's surface.
But more to the point, there's no evidence that the Dweller ships are actually made (predominantly) of metal. (Unlike the Mercatoria ships, whose descriptions do mention metal.)
Onboard the Poaflias:
They were on the foredeck, a slightly ramshackle sprouting of fibres and sheet protruding over the craft's battered-looking nose.
(Dwellers are not capable of surviving unprotected exposure to space: "And they do need to breathe. If the pressure is too low, they can't.")
So the answer is that the Dwellers built the ships they needed out of organic materials (carbon fibre is very strong, after all) with the smallest necessary amounts of metal gleaned from meteors.
Not all gas giant planets are structured like the ones in our solar system. Some have been found that have metals and water vapour in their atmospheres.
Here is an except from just one paper:
Transmission spectroscopy at higher resolution has yielded information on atmospheric dynamics, as subtle Doppler shifts of absorption lines can be used to assess the velocity and direction of atmospheric flow (Flowers et al., 2019; Miller-Ricci Kempton et al., 2012; Showman et al., 2013a). This has enabled planetary-scale wind determinations (Snellen et al., 2010), differential wind measurements between leading and trailing hemispheres (e.g., Louden & Wheatley, 2015), and constraints on the planetary rotation rate (Brogi et al., 2016). Ehrenreich et al. (2020) found evidence for strong (≈5.3 km s−1) day-night winds in the ultrahot Jupiter WASP-76b, along with evidence for condensation of iron on the planetary nightside due to a hemispheric asymmetry in its absorption signal. Recently, Tabernero et al. (2020) confirmed this detection of day-night winds for WASP-76b over a broader range of atomic and molecular signatures, including ionized calcium, atomic manganese, magnesium, sodium, potassium, and lithium. These are just a handful of the broad range of atomic and ionized metallic species and molecules that have been detected in hot and ultrahot Jupiter atmospheres with high-resolution transmission spectroscopy (e.g., Birkby et al., 2013, 2017; Brogi et al., 2014; Hoeijmakers et al., 2018, 2020; Nugroho et al., 2017; Piskorz et al., 2017; Wyttenbach et al., 2017; Seidel et al., 2019, for a recent review see Birkby, 2018).
It's also possible that space travel came to the original Dweller home world as a form of alien first contact.
In the end, I suspect that, unless someone finds a note on this from Banks, this question will be closed as opinion-based.