Where online can one find a database of convictions that allows one to
search convictions by statute section?
This doesn't exist.
Typically, the way the records a kept by the courts in the ordinary course will involve statistical categories that assign one category expressed in words to a case, that may encompass multiple statutory sections or subsections, and that don't capture secondary charges in a case.
For example, suppose that someone is charged with robbery, trespassing, burglary, theft, theft by use of a stolen credit card, and tax fraud. In all likelihood, in both most state court systems, and in the federal courts, this would be coded as a robbery cases for statistical purposes, regardless of the actual offenses of conviction. Similarly, the statistical record keeping process might not distinguish between half a dozen different fraud offenses on the books and simply categorize all such offenses as "fraud".
Worse yet, it would be common place for the statistical categories used by the judicial branch to be different from the statistical categories used by the department of corrections. In addition, each local jail would typically have its own separate statistical database for people held in its facility which would not have consistent categories with other databases in the state. At the federal level, the U.S. Sentencing Commission, has yet a third (and most detailed) approach to maintaining these records statistically for federal offenses. Also, it isn't uncommon for there to be more than one set of records maintained independently in a single state. This statistical information is typically reported in an agency annual report and sometimes in additional special reports or responses to legislative inquires (usually for budgeting purposes).
For example, in Colorado, criminal cases in state courts are maintained i a common database, but most municipal court ordinance violation cases are maintained in separate municipal court databases for each separate municipal court, with the exception of additional state databases for traffic offenses that count as "points" against your license handled by the DMV, and a separate state database where domestic violence offenses that disqualify someone from gun ownership are registered. In rural lower courts, the record keeping is often solely on paper or is in a computerized database that isn't even connected to the Internet. In urban areas and in federal courts, the records are kept electronically, but still aren't indexed or searchable in the manner you would like.
With the exception of the U.S. Sentencing Commission, only commonly charged crimes have statistical categories to assign them to at all, with most crimes on the books which are only rarely prosecuted simply lumped into "not elsewhere classified" or "miscellaneous".
In addition to those records, the U.S. Justice Department does two kinds of surveys each year which use different categories still. One it does directly in a random statistical survey with a large sample size of crime victims. The other is done through an annual survey of most (but not all) law enforcement agencies in the U.S. on a standard justice department form that gets filled out by some police department middle manager or administrator in each responding agency.
Murders are also kept track of in an independent vital statistics records maintained by coroner and forensic examiner's offices. U.S. immigration officials also keep a tally of deportable offenses reported to them and actions taken on those offenses.
The contents of the mittimus in a criminal case, which is the final summary statement of the precise offenses for which a criminal defendant was convicted, the sentence imposed for each offense of conviction, and the total sentence in light of which sentences are to be served consecutively and concurrently with each other, would generally not be included in a searchable database. There would be a databased from which you could pull the mittimus in a particular case for a particular offender, but the contents of the mittimus usually wouldn't be searchable. Maybe Oregon has better information systems than most, but I doubt it.
To get that kind of information, you'd need to do a freedom of information act request or the equivalent (and pay a state employee hefty hourly research fees), or have a legislative committee or administrative officer within the judicial branch or department of corrections authorize the search.
All of this information is a matter of public record (except in "sealed" cases). But gathering the information is very cumbersome.
There are a couple of social science databases maintained by particular academics, news organizations, or non-profit organizations, for particular purposes on a sporadic and incomplete basis that have some of this information, but the extent to which the manager of those databases will allow you to access that data varies wildly, and often that is also not complete.
For example, my small town hometown newspaper growing up, which published weekly, reported the offense of conviction (by name, not statute number) and the sentence imposed, for every conviction in the city's municipal court in the past week, with the name, age, and address of every convicted offender.
Often the prosecutor's office, the public defender's office, and the leading criminal defense lawyers in a particular county maintain private databases of this information that is more specific, and is almost never shared.