Title insurance companies could be a source; nearly all mortgages require a lender's title insurance policy, and knowledgeable buyers will insist on an owner's title insurance policy.
Real estate brokers usually maintain multiple listing services which can provide information on comparable properties on the market and closed listings (usually with sale data). MLS subscriptions would not be difficult to acquire for those in the business (lenders, title companies, appraisers, insurance adjusters).
Property tax assessments may provide a rough guideline, and perhaps general comparisons between neighborhoods and areas.
Most property and casualty insurance companies participate in insurance service bureaus, sharing information on policy limits and claims data. Insurance underwriters utilize weighted programs to estimate property values based upon factors such as location, fire protection district, construction type, square footage, features and amenities.
The main thing to keep in mind with appraisals is that it is only an opinion of value, and opinions are only as valid as the source. Also, the purpose for the appraisal needs to be considered; if there is a contract for sale the appraiser's estimate may vary from an appraiser for a mortgage loan refinance (a buyer may fall in love with a property, but a lender's primary concern will be risk exposure).
Edit to add: valuation may be determined in a number of ways. Replacement cost value may be much higher than actual market value based on recent sales of comparable properties. An income approach to value would be based on anticipated rental income revenue over a number of years (remaining useful life) plus land and salvage value. Zoning and permitted uses (as well as recent or anticipated changes) may have a profound effect on property value based upon a principle called "highest and best use" of the property.
All of the various approaches to value are likely to result in different numbers.