The PPI should have nothing to do with image quality unless you re-sampled and somehow reduced the total number of pixels available to a really low number. Don't worry about PPI unless the client wants a specific setting. Worry about total pixels in the image. Web Photos are generally in sRGB color space. How the client's software may compress your images for web display is beyond your control.

To get a good quality image occupying a full screen without distortion on most hi-res monitors you need a minimum height of 1080 pixels (2160 pixels for 4k monitors). If that is what the client wants I would use the following "File Settings" and "Image Sizing" on export:

Rather than quality, try the Image Sizing setting in the LR export. Restore the Quality setting to 100% in File Settings. Then select Resize to Fit "long edge" and check Don't Enlarge. Set the Long Edge to 2048 in pixels. Set the resolution to 132 pixels per inch. In section Output Sharpening, set for screen and Low (or Normal) depending on the amount of sharpening applied already during your processing. Depending on the online use, you might lower the long side to 1024 (or still smaller) and the PPI down to 72. Can this client provide any minimum requirements to identify if my suggested settings support or overshoot their minimum requirements? HighTail or email attachments should have no impact on the contents of the files being transferred, although in email the file must be attached rather than embedded.

Also, ask them what they've done to the files on their end? Used as-is or other processing such as down-sizing using their own techniques?