Why won't my vcr insert tapes?

Assuming there is power going to the machine, and the display is illuminated (showing clock for example), and the deck is empty, it sounds like a mechanical fault - could be worn belts or gears, but could also be a faulty/dirty mode switch (this can cause a lot of issues including failure to eject/load) - so tackle the simplest problem first - look at the belts and/or gears, especially those on the carriage/cassette cradle assembly; see if any are worn or broken.
If any belts are stretched or any gears look cracked (manufacturers used plastic gears which are often weak), see if you can order replacements. If those check out OK, next step is to examine the condition of the mode switch. To get access to this: unplug the VCR from the power point, and carefully open it up and locate the mode switch (usually UNDER the deck) and spray a little bit of CRC-2 electronic switch cleaner into the switch cavity (if the switch is a rotary type, gently rotate it in both directions to work the CRC cleaner into it - don't use too much - the cleaner evaporates but it will clean the contacts inside the switch and leave a fine lubricant to minimize wear). You do not need to open the switch or remove it. Then carefully re-fit the deck, connect everything (heads etc.) You can purchase CRC electronic cleaner from most stores (this is usually a small aerosol can with a hose attachment - ideal for switches and contacts). Then plug VCR in and test (use an old tape which has nothing important on it). If it still won't accept tapes, it's likely to be a mechanical alignment issue which requires a technician to correct as there may be broken or worn gears. (generally you cannot replace these efficiently or reliably by yourself without some experience, a gear/belt kit and a service manual to help you.)

*** I DO recommend you look at the belts and gears to check their condition FIRST before disassembling the machine or going any further. ***

Advice: being a modern design VCR, it's likely to have lots of flimsy plastic gears and other parts which break easily and may not be replaceable.