Grammarly for Safari

Thanks for the additional information. I would try installing Grammarly from here:

https://safari-extensions.apple.com/details/?id=com.grammarly.spellchecker.extension-W8F64X92K3

If that doesn't work, try to install another extension from there and see if that works (as a test).

Try First Aid:

Repair the Drive for El Capitan or Later

1. Restart the computer. Immediately, at or before thechime hold down the Command and R keys until the Utility Menu appears.
2. Select Disk Utility and press the Continue button.
3. Then select the indented (usually, Macintosh HD) volume entry from the side list. 
4. Click on the First Aid tab in the toolbar and wait for the Done button to activate. Click on it, then quit Disk Utility.
5. Select Restart from the Apple menu.

Download the extension here.

We can't be of any help until you tell us what error you received. From where are you getting extension?

Please read Writing an effective Apple Support Communities question. We are here to help. We are all users. We are not mind-readers nor clairvoyants. We only know what you write down and tell us. Be specific and thorough. Include your hardware information and installed version of macOS.

Is this a personal MacBook Air or something you have through work/school/etc.?

If it's a work/school owned MacBook Air you may have Mobile Device Management forced on your MacBook blocking any Safari extensions.

Is your account type Admin, or are you Standard or Managed? This could also block the installing of extensions in Safari.

Finally, do you have security/protection software that could be blocking your installing of Safari extensions? If so you would have to temporarily disable this.