Standard continuous wavelets behave like continuous bandpass filters. Since they rely on two continuous parameters (scale and shift), their strict implementation would be impossible, even on discrete data. It would require an non-countable numbers of convolutions(integral).
Then, the full story is much more complex and intricated, but some milestones follow.
First, people like Ingrid Daubechies have succeeded at sampling both the scale and the shift parameters to get a "discrete" and perfect implementation, that could be inverted. The subsampling factors, and the associated redundancy depends on the wavelet, and relate scale and shift. This can be send as a sampling theorem perform on the space of wavelet coefficients, just as the discrete Fourier transform samples regular frequencies in the frequency dmoain.
Then Yves Meyer managed to find an special wavelet family that allowed to make the transformation orthogonal and discrete, just like the discrete Fourier transform. And one knows that discrete orthogonal transforms, like the DFT, implemented in matrix form, can benefit fast algorithms (FFT) by clever relationships between Fourier coefficients at discrete frequencies.
Then, Stéphane Mallat among others managed to formulate a multiresolution analysis as a set of embedded subspaces with inclusion properties, yeilding a relation between two-sscale of father functions, or a father function and a mother fucntion. This linear relation could be interpreted as a convolution by a given filter. And the relation was valid across scales, and allowed a fast (linear in complexity) algorithm.
Finally, works have been performed, either to link those coefficients with specific wavelets. This allowed to derive which wavelets could be implemented as filter banks. The theory built on finite impulse response filters (finite support wavelets like Daubechies, Coiflets, Symmlets, etc.), multi-band (M-band) filters, but also to infinite impulse filters, recursive or redundant ones, even to nonlinear filters. The wavelet arises as the result of an infinite iteration of a basic filter-bank. Which one rarely does in practice, since most data are of finite length, with a limited number of practically useful scales.
So not all wavelets can be implemeted perfectly (invertible) with efficient filter banks. It is well known that discrete orthogonal wavelets cannot be real/symmetric/finite-length, except the Haar wavelet. Yet there exists a great deal of discrete wavelets that can be effciient on several signal/image/point cloud/mesh data.