Hi @chmarksiv!
As you weren’t extremely specific about a few details, I’m going to have work with what you’ve said so far and hopefully guess correctly on what you’re looking for. A quick disclaimer: I’ve never needed or used anything similar to what you are describing. I do love research though, so hopefully what I’ve discovered fits your needs.
Based on the information I have, I’d recommend ShootProof. It has a lot of fantastic features, and allows you to create custom online galleries for each client, offering them the ability to download, share, and view their photos easily. Of special interest to photographers who use Lightroom: ShootProof has a plugin that allows you to manage these galleries from Lightroom, and if you so desire, you can also integrate ShootProof galleries into WordPress sites with their plugin.
One VERY special feature that I do love: Web Apps. … It’s hard to explain without getting either really technical or geeky, but it functions as a native app without having to go through App Stores, doesn’t require downloading or installing, and works really well. The only drawback is that you can’t download or share photos through the app. Instead, you can share the “app’s download link”. Not perfect for you, maybe, but this is an optional feature. For a demo, click this link on an Android or iOS device: Sample App
ShootProof offers a variety of plans to suit different people’s needs, and has both a 30 day trial as well as a free plan with storage for only 100 photos.
I haven’t used ShootProof personally, but it is well reviewed and seems to offer what you were asking for…