Like published histories and Ancestry Public trees, DAR/SAR databases are compiled sources. They may be useful as guides, but the question I always have is what primary sources did the compilers draw from when that entry was created?
If you have not already done so, I recommend that you write out a biographical sketch and timeline for Vestus and his family, along with a list of all the records you have collected about him so far. Sibling records can be especially useful for filling in gaps in the family timeline and providing clues to migration.
The next step is to make a checklist of what specific primary records you might be able to find to verify the events in his life from your biographical sketch.
One useful checklist is Sources of Genealogical Information. So far you have been searching for an official birth record. Consider branching out to other types of records as well.
The Ohio Hypothesis
The FamilySearch Wiki article How to Find Ohio Birth Records is rough and needs a lot of work. However, it has this pointer:
In 1803, Ohio was created from Ohio Territory. Statewide registration
of births began in 1908 in Ohio. Records before 1908 are found in the
Probate Court in the county where the birth occurred.
Another factor to consider is coverage. It's not uncommon for coverage to be incomplete in the early years immediately after statewide registration is imposed, so even after 1908, it may be easier to find records searching at the county level. This means that if you don't have a good guess as to what county the family might have lived in, or where Vestus' mother might have been visiting at the time the birth happened, it might be worthwhile to back up and look for more records about the family later on, to find more clues.
The Pennsylvania Hypothesis
The FamilySearch Research Wiki's article on Pennsylvania Vital Records says that statewide registration began in 1906, so it is even more imperative that you locate what part of the state the family might have lived in, in order to pursue local records.
Once you have narrowed down possible locations for this family, see the question How can I determine what records are available in a particular locale? for more ideas about how to find records.
Another thing to consider: for effective searching, it helps to understand the nature of primary records and to learn about what you can expect to find out from them. For Ancestry.com, one way to learn more is to go into the Card Catalog and look at each individual record set. In the "about the database" notes on U.S., Quaker Meeting Records, 1681-1994 Ancestry says:
What Can I Expect to Find in Quaker Records?
There are generally two types of monthly meeting records, minutes
taken during the business meeting, and separate registers of births,
marriages and burials. Later in the 19th century many meetings began
to keep membership registers which incorporated more comprehensive
information. Each meeting kept records for individuals and families as
long as they remained faithful members of the meeting and within its
geographical boundaries. When a family moved from one meeting to
another, a letter (certificate of removal) was sent to the new monthly
meeting they would be attending. Notice of the transfer was written
into the minutes of the original meeting, and was also noted as
received in the new meeting. (emphasis mine)
Meeting minutes contain a recording of all business conducted in the
meeting. These include approvals of marriage intentions, records of
discipline, disownment, requests for burial in the meeting burial
grounds, and removal. Monthly meeting minutes rarely include
information about births and deaths.
You may not be able to find information about births and deaths in these records, but a certificate of removal and the family's arrival in the new area could be an important clue that would lead to other records. Note that not all names in this kind of record will be indexed, or indexed correctly, so browsing the images often yields far more information than you can find in any index.
Ancestry.com has 17 databases in the card catalog specifically pertaining to Fairfield County, Ohio. All seventeen of these are categorized as "Newspapers and Publications" or "Stories, Memories & Histories", which shows that Ancestry.com doesn't have any official records on the county level, only those which were collected statewide. So it might be more productive to look at research guides like the FamilySearch Wiki article on Fairfield County, Ohio or the Ancestry's Wiki from The Source on Fairfield County, Ohio. Examine any record collection you search, not just for the obvious question "does this database have my ancestors or not?" but also to understand the nature of the records, and to look for what clues it might give you to other records.
If you find your family in abstracted records (on Ancestry, via Google Books, Hathi Trust, Internet Archive, etc), try to get a copy of the original records if you can -- they may have more information than what appears in the published abstracts. Thomas C. Hill's work Monthly Meetings in North America: A Quaker Index has been made into an online searchable index -- see www.quakermeetings.com An example of an entry from the print version can be seen in the FamilySearch Research Wiki article U.S. Quaker Research (Society of Friends).