The LAN network in our office allows WiFi, but I am unable to download anything from Google Play. However, I am able to browse and visit various sites. Is there any way to get downloads, too?
Consult with the IT expert of you office and tell him/her your problem. He/She will then give you a best possible solution. If your office wifi network prohibits download then the IT expert may make an arrangement to help you in downloading necessary files.
Are you the network administrator? Do you have a proxy server or any server managed by someone else? Check if there's no blocking on downloads and WHAT types of download you're talking about. If you're trying to use P2P or torrent clients, it might be a block on your network and that can get you fired depending on the company policy. Hope it helps.
Check with your office IT guy, sounds like your phone is being blocked by the firewall possibly.
Also, can you download on your cell network? you may want to check your download settings on the phone. It might be as simple as enabling downloads over Wi-Fi.
What type of phone? What type of WiFi network? are you in a secured Network or just on public WiFi, like at a coffee house?
Talk to the administrator to find out if there are limitations on LAN network in your office not be able to download files. There are other android markets that allows to download apk to Windows desktop so from there you can add drag and drop with USB cable to android phone