I'm not a teacher. My grammar is fine, I know the basics well and with the right resources could probably teach it well.
But I make mistakes all the time in my texts and often in emails. Because I'm busy and don't always have the time to spell check. It's not the end of the world surely?
It depends what you are busy doing. If a large part of your job requires you to teach good spelling and/or grammar - or to demonstrate it on behalf of your company - then it is important. It's a large part of what you should be busy WITH.
If I were looking for somebody to build a wall for me and I found a website showing a gallery of their very good work, accompanied by lots of excellent independent reviews, which said "No job two smal - whatever you're requirments, get in touch with ourselfs and we garantee too build the best wall u could of dreamed have - at supprisingly low prizes", I would cringe slightly, but it wouldn't put me off using them. I would get the wall I wanted, at a good price; and if the builder is teaching anybody else, he will be teaching them to build good walls and not how to write well.
If an office-based company had a website like that - especially where they might have online/written influence on others and perceivably represent me - I wouldn't consider using them for a moment.
Also, what kind of software are you using in your emails? These days, separately running spell-check is largely a thing of the past, as it does it automatically as you type. Even MN does it - as with the 'sample' I wrote above, which is littered with red underlines. It's not just a case of forgetting to run spell-check; you have to deliberately ignore all of the red lines that you see before sending it anyway.
I would expect somebody who really struggled with SPaG to maybe just find it overwhelming and ignore all the red lines, but not somebody who does understand SPaG, but makes a few typos or has the odd brain fart, like we all do, and yet still chooses to ignore the red lines.