feat(vercel-edge): Add logs export

[AbhiPrasad]

Similar to #16165

Ensures we have logs support for the Next.js Edge Runtime.

resolves #16151

[github-actions]

size-limit report 📦

⚠️ Warning: Base artifact is not the latest one, because the latest workflow run is not done yet. This may lead to incorrect results. Try to re-run all tests to get up to date results.

Path	Size	% Change	Change
@sentry/browser	23.32 KB	+0.16%	+37 B 🔺
@sentry/browser - with treeshaking flags	23.13 KB	+0.17%	+38 B 🔺
@sentry/browser (incl. Tracing)	36.98 KB	+0.16%	+57 B 🔺
@sentry/browser (incl. Tracing, Replay)	74.15 KB	+0.07%	+50 B 🔺
@sentry/browser (incl. Tracing, Replay) - with treeshaking flags	67.49 KB	+0.04%	+24 B 🔺
@sentry/browser (incl. Tracing, Replay with Canvas)	78.8 KB	+0.05%	+39 B 🔺
@sentry/browser (incl. Tracing, Replay, Feedback)	90.6 KB	+0.03%	+20 B 🔺
@sentry/browser (incl. Feedback)	39.72 KB	+0.12%	+46 B 🔺
@sentry/browser (incl. sendFeedback)	27.94 KB	+0.16%	+43 B 🔺
@sentry/browser (incl. FeedbackAsync)	32.71 KB	+0.12%	+38 B 🔺
@sentry/react	25.12 KB	+0.13%	+31 B 🔺
@sentry/react (incl. Tracing)	38.95 KB	+0.06%	+21 B 🔺
@sentry/vue	27.55 KB	+0.16%	+45 B 🔺
@sentry/vue (incl. Tracing)	38.73 KB	+0.12%	+45 B 🔺
@sentry/svelte	23.34 KB	+0.14%	+33 B 🔺
CDN Bundle	24.52 KB	+0.1%	+25 B 🔺
CDN Bundle (incl. Tracing)	36.99 KB	+0.08%	+28 B 🔺
CDN Bundle (incl. Tracing, Replay)	72.03 KB	+0.05%	+32 B 🔺
CDN Bundle (incl. Tracing, Replay, Feedback)	77.35 KB	+0.07%	+51 B 🔺
CDN Bundle - uncompressed	71.57 KB	+0.11%	+79 B 🔺
CDN Bundle (incl. Tracing) - uncompressed	109.41 KB	+0.08%	+79 B 🔺
CDN Bundle (incl. Tracing, Replay) - uncompressed	220.69 KB	+0.04%	+79 B 🔺
CDN Bundle (incl. Tracing, Replay, Feedback) - uncompressed	233.23 KB	+0.04%	+79 B 🔺
@sentry/nextjs (client)	40.55 KB	+0.11%	+43 B 🔺
@sentry/sveltekit (client)	37.45 KB	+0.1%	+35 B 🔺
@sentry/node	143.45 KB	+0.04%	+51 B 🔺
@sentry/node - without tracing	96.53 KB	+0.05%	+44 B 🔺
@sentry/aws-serverless	120.86 KB	+0.04%	+47 B 🔺

View base workflow run

[chargome]

Looks good 🚀