Poison for Weapons, Tavern Brawler for Unarmed Strikes
Poisons
For slashing and piering weapons (the blowgun is a piercing weapon), you can use poison. Basic Poison is on the equipment list, and described on page 153 PHB:
Poison, Basic. You can use the poison in this vial to coat one slashing or piercing weapon or up to three pieces of ammunition. Applying the poison takes an action. A creature hit by the poisoned weapon or ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage. Once applied, the poison retains potency for 1 minute before drying.
It is not very economic, and the action to apply is a real cost here, but it can improve the damage without magic. Check with your DM, as not all DMs allow buying poison, some also make it illegal to buy in game and have in-game consequences; you can also create basic poison in other ways, such as with a Poisoner's Kit.
I'm not aware of a straightforward method that would work for the net. Nets don't deal zero damage, they don't deal damage at all, so you cannot add extra damage. You possibly could coat the parts of it that you will not touch in contact poison, for example with Oil of Taggit or Carrion Crawler Mucus (DMG, p. 257), but these poisons do not deal damage; for these kinds of poison it is also entirely up to your DM if they are available for sale.
Tavern Brawler
You can use the Tavern Brawler feat to improve the damage output of your unarmed strikes1 without magic and without Martial Arts (p. 170 PHB):
- Your unarmed strike uses a d4 for damage.
1 Note that unarmed strikes are not weapons but count as weapon attacks, as explained in the Sage Advice Compendium