How do I fix side button not working for app downloads on iPhone?

Try as suggested in this similar question:

Double click side bar for Apps - Apple Community

Go to Settings > Accessibility > Side Button

Change the Click Speed to Slow.

If the issue persists, install the mobile Apple Support app (if available where you live) and contact Apple using that. You can then optionally allow Apple technicians to have direct access to diagnostic data from your device. Alternatively, take or send it in for diagnosis:

Go to this page to find your nearest Apple Authorized Service Provider (AASP) or Apple Authorized Distributor (AAD) --> https://locate.apple.com/country

How to make a Genius Bar appointment

1. Go to Genius Bar Reservation and Apple Support Options - Apple

2. Go to "Get hardware help".

3. Select your hardware and continue selecting options requiring replacing parts until you see "Bring in for Repair"

4. Sign in with your Apple ID

or:

Refer to this document for ways to contact Apple ➞ Choose your country or region - Official Apple Support

Select your country (also look for "other" regions), then a product. If you don't see one that handles your issue then keep experimenting with selections until you reach one that gets you a chat session or a telephone call and get the representative to redirect you.

or:

Contact Apple for support and service by telephone ➞ Contact Apple Support - Apple Support

or:

For the iPhone Service Answer Center click here --> Choose Your Country or Region - Apple Support

Select your country.

For a cost estimate:

- Scroll down to "Get an estimate"

- Enter the service required

- Enter the iPhone model you have

- Enter the version of that model

- Click "Get estimate"

For instructions on how to get your iPhone, iPad, or iPod touch ready for service click here --> Prepare your iPhone or iPad for a service appointment - Apple Support

- "To protect your information and save time, follow these steps before you get service for your device."