Features

Adds a button to the browser toolbar. When clicked, it sends the URL of the current tab to Archive Today to preserve a snapshot of the page, and opens the result in a new tab.
Also includes Link context menu to Archive or Search, Page context menu to Search.
Archive URL can be bookmarked for future reference, shared, whatever. Uses HTTPS for security.
For similar functionality on Android, check out Archive Page in the Play Store.

Consent:

By using this extension you Consent to sharing your Archive or Search URL with Archive Today when and only when initiated by you. If you do not consent, remove and do not use this extension, which does not otherwise collect or share any user data.
Privacy Policy of Archive Today (as of this writing): Source IP is not disclosed, "but take in mind that when you archive a page, your IP is being sent to the the website you archive as though you are using a proxy (in X-Forwarded-For header). This feature allows websites (e.g shops or the sites with weather forecast) target your region, not mine."

Notes:

If you get an error, try again or try later. It's NOT an extension issue—Archive Today sometimes gets overwhelmed.
If you experience DNS errors, try a different DNS service (e.g., Quad9) or VPN.

What's New:

1.2.2 Update: Minor wording polishing. No functional changes.
1.2.1 Update: Ability to select an alias of archive.today for connection issues. Revamped onboarding/upboarding/options dialog.
0.9.0 Update: Remove (uninstall) button now in Options dialog. Update Notification only if enabled by user (optional permission).
0.8.4 Update: Add Privacy Policy to Options dialog. Display Options dialog when onboarding. Notify with extension is updated. Tweak default Options.
0.6 Update: Search logic improved.
0.5 Update: Option to open Archive Today in adjacent tab, tab at end, or active tab. Closing new tab now returns to original tab.
0.4 Update: Options to control new tab activation (separate for each function). New tab is activated by default. Uncheck option to not activate.
0.3 Update: Adds context menu items for: Page: Search Archive Today for the page URL Link: Archive or Search link in Archive Today

(Archive Today is a time capsule for web pages. It takes a 'snapshot' of a webpage that will always be online even if the original page disappears. It saves a text and a graphical copy of the page for better accuracy and provides a short and reliable link to an unalterable record of the web page.) Disclaimers:

Use at your own risk. No liability is accepted for any loss or damage.
This extension is not affiliated with Archive Today.
Trademarks are the property of their respective owners.

Developer response