I figured out a hack to be able to zap with any lightning wallet you want in the primal web app. Open up the dev console and paste this code:

window.webln.sendPayment = (invoice) => Promise.resolve(console.log(invoice))

Then whenever you click the zap button it prints the lightning invoice for the zap in the dev console instead of opening Alby. Copy and paste the invoice into whatever lightning wallet you want.

You can even save it as a JavaScript bookmarklet. Use this as the URL:

javascript:(function(){window.webln.sendPayment=(invoice)=>Promise.resolve(console.log(invoice))})();