Most rescues say No to this for very good reasons. Greyhounds in particular have spent all their working lives in kennels surrounded by other greys, so find it especially hard when they have to live on their own. This is why they often do much better in a home with two hounds - is this something you'd consider?
It is possible to combine dog ownership with full time working (though NOT a puppy). A rescue will want to see that you've got good arrangements in place. This could be a permutation/combination of doggy day care, dog walkers, family members etc. Bear in mind that a professional dog walker or doggy day care can be quite pricey so be sure you can afford it. Just saying vaguely "my dad could pop in once or twice a week" isn't enough - a good homecheck will want to see more detail and be clear that you've got reliable arrangements in place.
Please don't think that any home is better than a kennel - most of the greyhound rescues active in S Wales have a lot of their hounds in foster homes or a very nice sanctuary, in the case of Greyhound Rescue Wales. A dog that is placed in an unsuitable home will exhibit SA, will be stressed, may bark and will often end up as a bounce back to the rescue - this time even harder to rehome as it will now be stressed and unsettled.
You have a good choice of rescues. Greyhound Rescue Wales have a summer Fayre coming up at their sanctuary, which is in Ammanford on July 20th - more info on their website/FB page. That would be an ideal opportunity to see the conditions in which the hounds live before adoption (luxurious kennels with beautiful single beds, cosy duvets, heat lamps, lovely gardens and all manner of nice things). There will also be people on hand to chat to, hounds to meet - a nice day out and a no -pressure way to find out more.
You could also try Woofles who mostly take dogs from the last remaining track in Wales up at Ystrad Mynach - their kennels are near Tredegar.
There's also Rescued Racers in Swansea - a small charity who mainly bring dogs over from Ireland.
There's also Greyhound Welfare - the S Wales branch of the RGT. Call
01633 892846 - they are based near Rogerstone near Newport, so will be happy to home a dog in Cardiff. They run a monthly greyhound walk at Cosmeston Country Park at 2pm on the first sunday of hte month - meet in the car park. Again, a great way to come and find out more - have a stroll with the hounds, chat to dog owners.
S wales also has a Sighthound Walks and Playdates group on FB - we meet regularly for walks, sighthound playdates, sighthound agility etc. Relaxed, sociable gatherings, usually at indoor riding schools.