I found this a challenge for looking up this family.I see you couldn't find anything on Joseph Ward or Alice Ward in Hoopeston in Vermillion co., Illinois. I couldn't find anything concrete for Joseph, but I saw on you member Tree you had his father and mother.
As for Alice Ward, I might have found a good candidate.
In looking at at Censuses many the middle name is used
like Alice, and many time birthdates are wrong.
I found in Hoopeston in the 1900 Census, a May A Ward
born 1871, Indiana. She was 29 years old, and her mother May C Barret was 60 years old, and widowed. This May was listed as married, (even though no husband was listed), she had 1 child, and it wasn't listed, she was also married 2 years.
I saw on that 1910 Census that Mabel Ward was born in 1897.
Like you I couldn't find a Joseph for Hoopeston in 1900.
This May C Barret was from Stateline which is in Warren county, Indiana which is right over the state border of Hoopeston, Illinois. She was born in New York, not Indiana
like the 1910 Census you found for Alice's mother.
In 1880 this May was Mary C Barret, and her daughter was Mary Barret.
It is a good possility since there were not many Ward born in Indiana, in 1871 and lived in Hoopeston, Illinois.