Short version: open the free tool, drop your file, download the result. Nothing is uploaded.

Steps

Save the PDFs to the Files app first (from Mail or Messages: open the PDF, tap Share → Save to Files).
In Safari, open WBS PDF Merge and tap the box, then Choose Files.
Pick the PDFs (tap Select in the top corner to choose several), then fix the order with the ↑ ↓ arrows.
Tap Merge PDFs, then Download. The combined PDF lands in Files → Downloads.

Why this way?

Many "PDF merger" apps are free to install and then ask for a weekly subscription or add a watermark. Many websites upload your documents to merge them. WBS PDF Merge runs inside Safari, so a lease, a payslip or a medical form never leaves the phone, and there's no watermark. Free merges take up to 20 PDFs at once.

The built-in option: Create PDF in Files

Open the Files app and go to the folder with the PDFs.
Tap ⋯ → Select and tap each PDF.
Tap ⋯ at the bottom and choose Create PDF.

It works without any website, but the pages go in the order you tapped the files and you can't reorder them afterwards. If the order matters, rename the files first (for example 1 lease, 2 addendum) or use the Safari method.

Troubleshooting

Can't find the downloaded file? Open Files → Browse → On My iPhone (or iCloud Drive) → Downloads.
A PDF is password-protected? Remove the password first; locked PDFs can't be merged.
Want to add photos too? Turn them into a PDF with JPG to PDF, then merge.

The same steps work on Android in Chrome: the merged PDF goes to the Downloads folder. On a computer, see how to merge PDF files without uploading them.

More free guides

All tools: WBS Tools · All apps: Wood Between Studio apps