When I joined my current team, I noticed that we consumed a lot of canned/bottled drinks and had no recycling bins. I asked the office manager (who's responsible for the whole floor, not just us), who told me that we used to have them but people misused them and it was too much trouble, so they stopped. I didn't ask to change that; I was just gathering information.
Next, I investigated whether recycling could be effectively collected. One night when I was staying late enough, I saw the cleaning person and asked if they separated recycling -- if we got a bin for our team, would it end up in recycling or would it just go into the trash anyway? The cleaner told me they collected recycling (and belatedly I noticed the second can on his rolling cart). So I went to the store, bought a bin, put it in our team's common room, and told my team-mates to not make a mess.
I told this story for two reasons. First, your first step should be to gather information. Bins are easy; it's what happens to their contents next that matters. Find out if the people who are already servicing your building are set up for recycling already -- and, if so, if it's just a matter of having bins, or if it would incur extra costs.
Second, armed with that information, approach the powers that be about implementing it. A busy office manager responsible for your whole building will react more favorably to "I checked and we're allowed to have recycling bins at no extra service cost; would it be ok if we got some?" than to "could you get us some recycling support?". Make it easy for the person who has to approve this to say "yes".