Answer: The short answer is "no," you probably do not need to provide shower facilities. However, there may be exceptions, so read through this entire answer to determine whether you may need to seek specific legal advice.
The Oregon Administrative Rules ("OAR") require RV parks to provide "toilets," but do not specify that parks are required to provide showers. OAR 333-031-0066 (1)(a).
The OARs refer to the types of "toilets" allowed, which can include flush toilet facilities, pit privies, or chemical toilets. OAR 333-031-0066 (2) and (3). Given that an RV park can provide non-plumbed toilet facilities, there is little doubt that showers are not part of the facility requirements.
Keep in mind that each situation is different and could be affected by local ordinances or your park's specific recreational park license or building permit. You should check your park documents for any mention of a shower requirement, and check your city or county ordinances governing RV parks for the same. If you are unsure whether shower facilities might be required, you should consult an attorney to specifically evaluate your situation.
(Footnote: If you do provide showers at your RV park, then the OARs lay out several specific requirements. Shower walls, ceilings and partitions must be impervious to water and watertight seals must be maintained at joints. Floor areas must be finished with non-slip surfaces and sloped to effectively drain water. Wooden racks or duck boards over shower floors are prohibited. Where glass shower doors are used, they must be made of safety glass. OAR 333-031-0012 (1).)