Answer. I believe that what you want to do is possible; you want to verify with your own legal counsel.
But let me suggest that you asked the question in the wrong way. Rather than asking whether the law allows this, you should ask whether the law disallows it. In other words, in most cases, when it comes to landlord-tenant legal rights and duties, Chapter 90 sets a “floor” below which you cannot go. So although the law does not address this specific issue, that does not mean you cannot do institute such a policy, so long as it does not violate other portions of the chapter.
What laws might apply? Here are the ones that come to mind:
90.130 Obligation of good faith. Every duty under this chapter and every act which must be performed as a condition precedent to the exercise of a right or remedy under this chapter imposes an obligation of good faith in its performance or enforcement. [Formerly 91.730]
90.135 Unconscionability. (1) If the court, as a matter of law, finds:
(a) A rental agreement or any provision thereof was unconscionable when made, the court may refuse to enforce the agreement, enforce the remainder of the agreement without the unconscionable provision, or limit the application of any unconscionable provision to avoid an unconscionable result; or
(b) A settlement in which a party waives or agrees to forgo a claim or right under this chapter or under a rental agreement was unconscionable when made, the court may refuse to enforce the settlement, enforce the remainder of the settlement without the unconscionable provision, or limit the application of any unconscionable provision to avoid an unconscionable result.
(2) If unconscionability is put into issue by a party or by the court upon its own motion the parties shall be afforded a reasonable opportunity to present evidence as to the setting, purpose and effect of the rental agreement or settlement to aid the court in making the determination. [Formerly 91.735]
90.220 Terms and conditions of rental agreement; smoking policy; rent obligation and payment. (1) A landlord and a tenant may include in a rental agreement terms and conditions not prohibited by this chapter or other rule of law including rent, term of the agreement and other provisions governing the rights and obligations of the parties. ****
90.262 Use and occupancy rules and regulations; adoption; enforceability; restrictions. (1) A landlord, from time to time, may adopt a rule or regulation, however described, concerning the tenant’s use and occupancy of the premises. It is enforceable against the tenant only if:
(a) Its purpose is to promote the convenience, safety or welfare of the tenants in the premises, preserve the landlord’s property from abusive use, or make a fair distribution of services and facilities held out for the tenants generally;
(b) It is reasonably related to the purpose for which it is adopted;
(c) It applies to all tenants in the premises in a fair manner;
(d) It is sufficiently explicit in its prohibition, direction or limitation of the tenant’s conduct to fairly inform the tenant of what the tenant must or must not do to comply;
(e) It is not for the purpose of evading the obligations of the landlord; and
(f) The tenant has written notice of it at the time the tenant enters into the rental agreement, or when it is adopted.
(2) If a rule or regulation adopted after the tenant enters into the rental agreement works a substantial modification of the bargain, it is not valid unless the tenant consents to it in writing. ****
90.510 Statement of policy; rental agreement; rules and regulations; remedies. ***
(6) Every landlord who rents a space for a manufactured dwelling or floating home shall provide rules and regulations concerning the tenant’s use and occupancy of the premises. A violation of the rules and regulations may be cause for termination of a rental agreement. However, this subsection does not create a presumption that all rules and regulations are identical for all tenants at all times. A rule or regulation shall be enforceable against the tenant only if:
(a) The rule or regulation:
(A) Promotes the convenience, safety or welfare of the tenants;
(B) Preserves the landlord’s property from abusive use; or
(C) Makes a fair distribution of services and facilities held out for the general use of the tenants.
(b) The rule or regulation:
(A) Is reasonably related to the purpose for which it is adopted and is reasonably applied;
(B) Is sufficiently explicit in its prohibition, direction or limitation of the tenant’s conduct to fairly inform the tenant of what the tenant shall do or may not do to comply; and
(C) Is not for the purpose of evading the obligations of the landlord. ****
Conclusion. Note that ORS 90.295 pertains to “applicant” screening charges, etc. The statute is very specific and detailed. However, that is because it is limited to situations where a prospective tenant is seeking entry into the park as a tenant. There is nothing in Chapter 90 dealing with background check for guests who will remain at the park in excess of 14 days. However, if I were to draft a rule to implement your suggested policy, I would likely use it as a general template of issues that should be addressed, since some of them are the same, e.g. denial of the right to remain at the premises because of a failure to pass the background check. Additionally, you would want to make the list of disqualifying criteria known in advance, so tenants and applicants knew whether it would be worth the cost and effort to apply.
Lastly, I suggest that you implement such a rule through a rule change in accordance with ORS 90.610. Good luck!
 For example, on nonpayment of rent notices, you would not want to institute a payment period shorter than 72 hours. Similarly, you would not shorten the cure time for a rules violation to less than 30 days. Conversely, if you wanted to lengthen the time to pay late rent or cure a violation, you can certainly do so.