Some species root easily from thicker sticks and some may even root after shipping but the vast majority of desirable species for bonsai are a just a touch more challenging. I doubt very much that there will be many reliable sellers specialising in cutting material by mail though as there will be a very small market of people who would trust mail order cuttings and had the skills to get the resulting sticks to root.
I would also be looking locally for cutting material because fresh is far better and more reliable.
It will also pay you to do some practice to hone skills and setup for getting roots to grow before starting off with more challenging species.
You may also have some challenges with quarantine. Not sure what protocols Ireland has for plant import but many places require certificates and permits or spending time in a quarantine facility.
Elms - from either stem cuttings or root cuttings. Root cuttings can give nice fat trunks with good character if you can get twisted roots.
Chaenomeles and other quinces. Chaenomeles (Japanese flowering quince) grows as a multi stem shrub in many gardens and rooted stems can be divided off the clump to save the time and stress of striking cuttings.
Azalea (probably easier to root in summer)
Callistemon and Melaleuca (also easier in summer)
Ficus - (most are frost sensitive so summer is better for cuttings)
Olive - but quite slow to grow in a pot. Most impressive olive bonsai are collected feral trees
Serissa - strikes very easily but also divide suckers from the clump that may already have roots.
Arbutus - 'Irish strawberry tree' - though I'm sure you guys have another name for it.
There are probably many others that you may find in neighboring gardens and parks.
It is a well known fact that cuttings taken without permission over the fence will root far better than any you have asked for. Anything hanging over or through a fence is fair game - Just kidding. Most gardeners are more than happy to give small sections of plants if you show an interest in their gardens.
Another, far better, source for bonsai material is feral weeds growing in abandoned lots or roadsides. Garden renovations are also a great source for established plants. Look for building renovations and ask the builders if you can have the plants they dig out to make way for the extensions. Offer to help friends, relations and work colleagues to remodel their garden. You get well grown plants instead of the tedious waiting for cuttings or seeds to grow and many species transplant far easier than we give them credit for.