How does VB theory explain the Si-O-Si bond angles in SiO2?
Tl;dr version: the O-Si bonds utilize sp hybrid orbitals so that “lone pairs” are in nearly pure p orbitals and, rather than being true lone pairs, engage in some sort of $\pi$ bonding, the exact nature of which is unknown.  Other hypotheses are that the bond is highly ionic or that it is favored due to oxygen-oxygen repulsion.
Long version: First, let me point out a slight inaccuracy in the posted question:

Since the oxygen bonds to two silicon atoms, one would expect a sp3 hybridization, with two orbitals filled with lone pairs —

This is an incorrect interpretation of VB theory. Although we might approximate a molecule such as $\ce{H2O}$ as having an sp3-hybridized O atom, quantitative approaches to VB theory account for the fact that the contribution of s and p atomic orbitals to localized orbitals differs if the group differs. That is, lone pairs and bond pairs can have (and usually do have) different contributions from the s and p orbitals, ie different hybridization. A commonly cited example of this is $\ce{SH2}$, where the S-H bond orbitals are nearly entirely comprised of p orbitals, so the lone pairs are necessarily essentially sp hybrids.
Furthermore, one quantitative approach calculates the p orbital contribution based on the bond angle (Coulson's Theorem). In silicate, the Si-O-Si bond angle varies between ~140$^\circ$ and 180$^\circ$, depending on polymorph. We can calculate that the hybridization of these bond orbitals thus varies from sp (180 degree angle) to sp1.3 (140 degree angle), which is actually a relatively small variation in the s orbital fraction (43-50%). The lone pairs, therefore, are nearly pure p (93-100%) orbitals.
Although it is known that the $\alpha$-quartz polymorph (bond angle 144 degrees) is the most stable at low temperature, the observation of many polymorphs suggests that there is not much of an energetic penalty for the variation in bond angle within this range.
For simplicity, let’s consider the extreme case of the 180 degree bond angle, so that the O-Si bonds are formed from sp orbitals on O, and the lone pairs are in pure p orbitals. Both VB theory and MO theory indicate that this arrangement is only energetically favorable relative to a bent conformation if either the p orbitals are empty (as in $\ce{BeH2}$ for example) or if the electrons in the p orbitals are involved in $\pi$ bonding, for example the carbon atom of $\ce{CO2}$ or $\ce{HCN}$.
COnsistent with this idea, researchers have argued since at least as far back as the 1960's that the “lone pairs” on oxygen in silicate are in fact not lone pairs, but are involved in $\pi$ bonding to the silicon. This is supported by the short Si-O bond lengths and high bond strength which suggest a bond order greater than 1.
Historically, this interaction has been proposed to involve empty d orbitals on silicon, but more recently there has been a shift away from d orbital involvement, for similar reasons that d orbital involvement in phosphorus bonding has generally been disregarded. Interaction with empty $\sigma^*$ antibonding orbitals on Si, however, also does not appear satisfactory, as that would weaken the Si-O bonds (assuming that all Si-O bonds are equivalent, so all would have increased bond order from $\pi$ bonding, but decreased from addition of density to $\sigma^*$). Essentially, there would be a significant contribution of a resonance structure in which each Si has two Si-O bonds and one Si=O bond, similar to a carbonate.
Nonetheless, the idea that a large bond angle is favored due to some sort of $\pi$ bonding seems to be the prevalent view. Two other hypotheses, however, are still supported by some.
Other hypotheses
Ionic interactions
One alternative explanation is that there is very little difference in bond orbitals and lone pairs, which could result from a strongly ionic structure of $\ce{O^2-}$ anions and $\ce{Si^4+}$ cations[1]. On the oxygen, all of the electrons then are lone pairs, and the “bond angle” (which isn’t between any true bonds) is simply optimized to minimize repulsion between negatively charged oxygen atoms and between the positively charged silicon atoms. This would result in a tetrahedral arrangement of the negative charges around Si (as is observed) and a linear arrangement of the two Si atoms around each oxygen (the occasionally observed 180 degree angle). Since there aren’t any bonds, VB theory doesn’t really apply, and we consider the oxyanion as a lone atom with completely filled valence shell.
The problem with this argument is that free $\ce{Si^4+}$ is not observed under any circumstances, and most computational approaches suggest a high degree of covalency in the Si-O bond.
Oxygen-oxygen repulsion
A third hypothesis is that VB theory does not apply here because the minimization of energy of the Si-O-Si bond angle is not the only factor. Specifically, oxygen-oxygen repulsion favors a larger angle so as to increase the distance between silicon atoms and, therefore, the oxygens attached to adjacent silicon atoms.
Even at the small scale of a single molecule of $\ce{(HO)3Si-O-Si(OH)3}$, it has been argued that oxygen-oxygen repulsion is sufficient to cause a large Si-O-Si bond angle without any need for $\pi$ bonding[2]. The observed angle is a compromise between repulsion, which favors a 180 degree angle, and bond orbital energy, which favors a bent structure with bond angle closer to 110 degrees.
A problem with this hypothesis is that the Si-O-Si bond angle in disiloxane ($\ce{H3Si-O-SiH3}$) is also observed to be 144 degrees, even though there should be no oxygen atom repulsion effects and general steric effects should be much less than in silicates.
[1]Gillespie, RJ and Johnson, SA (1997) Inorg Chem 36:3031-3039. doi:10.1021/ic961381d
[2]Noritake F (2019) J Comput Chem Jpn 5:1. doi:10.2477/jccjie.2018-0016