Genuine problem, not a rant

Started a new frontend developer/designer/graphic designer job recently. I feel technically capable, no problem there. Just noticing a pattern of repeatedly missing deadlines.

Its a very busy office, with various people all coming to me with things they need me to do. Never worked in an environment so busy before.

Doesn't help that they force me to manage my tasks by spreadsheets, communication by email, deployment by filezilla and no version control, but not convinced I'd still meet deadlines if I had a better setup.

In what way could I improve upon how I run things?

Could you elaborate on your past experience please?

You can show them how to do the things better.
-Use a VCS (Gitlab has free private unlimited repos)
-1 click deployment (Jenkins is my choice here)
-A tool for planning (Redmine is a good option, but you need a server. I don't now any free option that works "as a service")

Sadly, I think you should invest your personal time to improve things.
The company you describe is like the company I work for... And I did some of the changes mentioned here (and others were made by my colleagues).

Its a long bumpy road, and if it doesn't work: at least you learn a lot of things

Hmm, well I have really inherited a mess from my predecessor.

Sending several email marketing campaigns a week and he simply copied a huge HTML file and rewrote it every time. I've begun converting templates into Foundation for emails templates.

We host several websites for our own in house brands. They need to be updated by non technical types so a mixture of WordPress and Joomla is their preferred method. I've recently discovered Statamic which I will be building all their marketing sites on in the future if they're willing to spend the money on the licensing. If not I'll probably just quit and run.

My personal preference of VCS is git hosted by bitbucket. We're only a small dev team so that would be free to use, plus can hook in easily to Jira which would only be $10 self-hosted.

I just need to convince them to buy me a decent PC with PhpStorm and a version of Photoshop and Illustrator newer than CS4!