Can I upgrade the networking card
"the" networking card? I'm guessing you don't know there are two. Ethernet and Wireless.
The Ethernet adapter can't be upgraded or changed out. It is an integrated component. The Wireless adapter is indeed an actual card. 802.11 B/G to be more precise. In fact, there are two different possible wireless cards that Dell put in there... and since even THEY don't know which one ended up in the laptop, they include drivers for both on their support web site. Both are B/G cards.
The card itself is a full sized mini-pci card. That's one of them.
Can you upgrade it? Sure. As long as it is a full sized mini-pci card. Here's an N protocol card, max transmission speed of 300MBPS, instead of the 54MBPS that is currently in your laptop....
It would be better to buy a USB wireless network adapter and disable the internal one. You should be replacing the old 1525 soon.
I still have, and use regularly, a Dell Inspiron 1525 machine that I bought for my wife in summer 2009 for school use.
Ten years later... the machine is now running an Intel Core 2 Duo 2.4GHz T8300 CPU with 4GB RAM. The original Vista OS has been upgraded to Windows 10 Pro 64-bit, version 1903 (OS Build 18362.239). The machine connects to my LAN via an upgraded Intel Dual-Band Wireless AC-7260 WiFi adapter. LAN traffic speeds approach 750Mbps (on 5GHz band) across my Gigabit LAN using this WiFi adapter on my Inspiron 1525 machine - and Internet download speeds approach 450Mbps across my Gigabit Internet connection.
The Intel AC-7260 PCIe WiFi adapter is a 'half-height mini card' (HMC, E-keyed) board format that connects to PCIe x1 connectors. The Inspiron 1525 typically shipped with the Dell DW1390 or DW1505 'full height' PCIe WiFi adapters. So, you will need an HMC-to-full height mini card installation adapter to properly install the AC-7260 WiFi adapter in the Inspiron 1525.
The Inspiron 1525, configured as my machine is now, outperforms newer Dell XPS laptops that cost more than 2x the $$$ I have invested in this older Inspiron 1525 machine - so long as the intended machine use is not intense video gaming.
So, if you desire to update/upgrade the WiFi adapter in the Inspiron 1525 I can confirm that the Intel Dual Band Wireless AC-7260 WiFi adapter works correctly under Windows 10 using the Intel AC-7260 Win10 drivers.