Vanished is definitely the apt description for them Wendy. I tried all different methods with no luck in England or Wales. There are four possibilities - I am presuming they are direct ancestors of yours so the chance of the whole family having died is not likely.
So therefore:
1. The pages has not yet been transcribed or was accidentally missed
2. The page has been destroyed (I found some of my ancestors on a partially destroyed page on the 1871 census).
3. They somehow managed to get missed on census day.
4. They were off travelling somewhere else outside of England or Wales.
The search I did included the US census and nothing obvious came up on there, but they may have gone elsewhere such as Ireland, Canada or Australia. I have a couple of cases in my family where they went to a foreign country for a couple of years either with work (soldiers mostly) or to "try it out" but ended up coming back to the UK. I would presume if it was the later you would probably have a "family story" about it but if it was the former then you might not.
I did find the birth entry of:
Reynolds, Thomas Richard Llandilofawr 11a 783
but could not find any of the other children's birth entries.
I found the family on the 1881 census at Llandilofawr as no doubt you have and then William living with his parents, Elizabeth and Richard and brother Thomas, on the 1871 census with his birthplace spelt Alternon Cornwall. Again on the 1861 census with it spelt Altarnun, same parents and sibling and also aunt Mary Ann.
The son William may have died in 1909. There is a death record in Jan 1909 Qtr for a William James Reynolds registered at llanelly Carmarthenshire. There are also other Reynolds registered in this area that do not appear to be yours so this one may not be either.
Good luck hopefully you will find them when the 1911 census is released.
