If you are using Laserfiche Web Client's PDF Viewer, you may do so with a custom script. 
* This is tested on Chrome / Laserfiche Web Client 10.4 and may not work with other version without adjusting.

From Web Client Options, Add the following two Custom Actions

1- Name: PDFViewerZoom Save

Location: Document Viewer toolbar

Command:

var docViewerDoc = frames[3].document
var docViewerURL = docViewerDoc.location.href;
if (docViewerURL!='' && docViewerURL.indexOf('PdfViewer')>0) {
	var entryId = webAccessApi.getFocusedEntries()[0].id;
	var scale = frames[3].PDFViewerApplication.pdfViewer.currentScale;
	localStorage.setItem('Zoom-'+entryId, scale);
}

2- Name: PDFViewerZoom Load

Location:  Document Viewer toolbar

Command: 

var docViewerDoc = frames[3].document
var docViewerURL = docViewerDoc.location.href;
if (docViewerURL!='' && docViewerURL.indexOf('PdfViewer')>0) {
	var entryId = webAccessApi.getFocusedEntries()[0].id;
	var scale = localStorage.getItem('Zoom-'+entryId);
	frames[3].PDFViewerApplication.pdfViewer.currentScale = scale;
}

Enable script:

if (typeof top.alreadyZoomed == 'undefined') {
 top.alreadyZoomed = true;
setTimeout(function() {
var docViewerDoc = frames[3].document
var docViewerURL = docViewerDoc.location.href;
if (docViewerURL!='' && docViewerURL.indexOf('PdfViewer')>0) {
	var entryId = webAccessApi.getFocusedEntries()[0].id;
	var scale = localStorage.getItem('Zoom-'+entryId);
	frames[3].PDFViewerApplication.pdfViewer.currentScale = scale;
}
}, 2000); // auto zoom after 2 seconds, Adjust timer to accommodate network speed

}
return true;

After saving, move these custom actions from [Under More Actions button] to [On toolbar].  This script is going to save the Zoom setting when the user clicks the Save Button and associate it with that Entry Id. The setting will be saved in Browser's local storage. So it will reset if user change browser or clears local storage. 

This is a very simple script so modify it if you want more advance way of handling that, and in overall; Use this script as an idea on how that feature can be implemented.