Is it possible to initiate Steam trading without adding people as friends?

Share your third party site trade url with them.

Here is how I do it:

Go to 'Inventory' -> 'Trade offers' -> 'Who can send me offers?'

Here you will find the following box contains a sharable URL.

Another option is to use the steam trading forums. (this information is found on the same 'who can send me offers?' page)

There are two ways you can initiate an out-of-game trade with Steam. You have to either be friends with them or be in the same group as them.

In both cases, the process is roughly the same. For someone you are not friends with but are in a group with, open the friends list, go to the groups tab and find the group you share.

If you are already friends with someone, you can simply use the friends tab.

Then, locate the person you want to trade with, click the triangle next to their name and click Invite to Trade.

Finally, you can now also start Steam trades merely by virtue of being connected to the same TF2 server. The old eight-slot in-game TF2 trading system has been retired, and thus the Objects → Trading → This server screen now invokes Steam Trading.

If you're specifically looking to trade TF2 items for TF2 items, you can do so through TF2, without adding someone to your friends list, you just need their steam community url. To do so, go into:

Items > Trading > Steam Profile

If you're looking to use Steam to trade (instead of going inside TF2) then you will need to add them as friends.