Webview'da download yapabilmen için önce webview için download kütüphanesini tanıtmalısın. örnek:

webview.setDownloadListener(new DownloadListener() {
    public void onDownloadStart(String url, String userAgent, String contentDisposition, String mimetype, long contentLength) {

        Uri webpage = Uri.parse(url);

        DownloadManager.Request request = new DownloadManager.Request(webpage);
        request.setMimeType(mimetype);
        String cookies = CookieManager.getInstance().getCookie(url);
        request.addRequestHeader("cookie", cookies);
        request.addRequestHeader("User-Agent", userAgent);
        request.setDescription("Downloading file...");
        request.setTitle(URLUtil.guessFileName(url, contentDisposition,mimetype));
        request.allowScanningByMediaScanner();
        request.setNotificationVisibility(DownloadManager.Request.VISIBILITY_VISIBLE_NOTIFY_COMPLETED);
        request.setDestinationInExternalFilesDir(MainActivity.this,Environment.DIRECTORY_DOWNLOADS,".pdf");
        DownloadManager dm = (DownloadManager) getSystemService(DOWNLOAD_SERVICE);
        dm.enqueue(request);
        
        Toast.makeText(getApplicationContext(), "Downloading File",Toast.LENGTH_LONG).show();
    }
});

manifest dosyasına izinleri eklemeyi unutma. Kodu kendine göre dizayn edebilirsin.