Well the way I found my ancestors who were in orphanages was through the U.S. census, but I no longer have that subscription so you will have to search it if you have it.
Search U.S. census for Vermont in the year 1850, I would just search Sarah at first, keeping in mind the correct one will probably be in or around Charlotte, Vermont and that she will be 9-10 years old depending on if her birthday is before or after the cenus date. Once you check on one possible "hit" you will see what date the census was taken on top of the page and KNOW if she would have been 9 or 10 so when you chaeck future hits you can screen by the right age. I found one of my ancestors doing this.
If there were no orphanages in that city try Burlington and any other city that had one.
I'm pretty sure the 1840 census did NOT list dependents by name I THINK the 1850 did but it might not. If not you will need to get a list of orphanages in operation in those years and try and contact them each and ask them to search their records. Even if 1850 does list names you might find a list of orphanages helpful because it will tell you what cities to keep an eye out for when you search the census.
And of course keep an eye out for all those "maiden" names also. KEEP the FAITH! One problem I have run into time and time again is that so many records are not on the web yet. What you can NOT find today will only be one mouse click away 10 years from now, or maybe even next week. So even if you don't find it now check back often.