Did you ever get this information? If not, I finally made it to the Oregon State Library this week and made copies of the list. I'm afraid I have bad news: there are no Cunninghams listed on Sept 10, and no J Cunningham or any Cunningham with a large family on any date. Here are the Cunninghams that ARE listed:
Cuningham, E. - Sept 18;
Cunningham, (Mrs.) & son - Aug 20;
Cunningham, S. - Sept 12;
Cunningham, W.M. & wife, August 30.
Note that the list names single individuals or heads of households only, with just a count of family members, e.g. wife, son(s), daughter(s), so E. Cunningham on Sept 18 and S. Cunningham on Sept 12 were single. Female heads of households were consistently listed as (Miss) or (Mrs.).
What information do you have that your family should have passed through there on Sept 10, 1853? It's possible they did but was not enumerated - the introduction states that some trains were missed if the "unnamed enumerator" was absent or ill. If you are sure they arrived in 1853 (e.g. from DLC), perhaps they split off in Idaho and came via a southern route.
Also, this document is an index and there may be transcription errors. The original article was on page one of the Oregonian on October 29, 1853 - it may be possible to find it online. The Oregon Historical Society has a lot of early Oregonian microfilms, and perhaps Genealogy Bank or other archive sites would have it. I will look for it next time I am at OHS, whenever that is!
I just did a search on paper-trail.org - found James Cunningham listed on a survey of "Crossing the Plains and Mining in Oregon and Caliornia", a reminiscence by George Miller West. He is listed on the roster of a company of miners in 1852 - the Butler party. It also says "non-emigrant mention," which doesn't make sense if he was on the roster. This book is a memoir that covers quite a few years - including an 1853 trip - dates are by year only. It is at the Oregon Historical society, the Bancroft Library (UC Berkeley), and at the Nebraska State Historical Society.
Paper-trail.org also has a diary for 1853 that lists "Cunningham" - sounds like a miners' company too. They cut off at Soda Springs (Idaho) to Salt Lake City and thence via the Applegate Trail to Jacksonville OR, arriving 10/8/1853. Cunningham is mentioned on 9/27/1853. No details here either; these surveys are bare-bones extracts. The original is at the Newberry Library in Chicago - sounds like it is pretty detailed.
There are some other Cunninghams in paper-trail.org, all 1850 & miners rom the sound of it. In any case, these all sound like single men, miners, not a head of household traveling with a large family.
If you are interested I can send you pdf's of these - you have to have a paid subscription to OCTA to look at them on paper-trail.com. I can't attach them here because they are pdf's.