How much energy does a small spider expend per day just waiting for its web to vibrate?
Short answer
Approximately 240 J on a daily basis.  

Background
Ballesteros et al. (2018) modeled basal metabolic rates of insects. They reckoned that endotherms, like insects, basically use energy directly correlated to the number of cells, which is linearly correlated to their body mass. They checked their model with experimental data from Chown et al. (2007) (Fig. 1).

Acknowledging that arachnids aren't insects, but that both are arthropods, let's take an average-sized web making spider, like the black widow, weighing in at 1 g. This yields a basal metabolic rate of 10 J/h, or 240 J/day. However, and given the linear correlation between energy expenditure and mass, a dwarf spider weighing in at 1 mg expends 1000 times less, and a Goliath bird eater approaching 200 g will expend 200 times more.

To convert the unit for the Black widow to a Watt scale, we get 1 J/s that corresponds to 1 W, so 1 J/h corresponds to about 0.3 mW.


Fig. 1. Base metabolic rate for insects. Data from more than 300 species were extracted from Chown et al. (2007). source: Ballesteros et al (2007) 

References
- Ballesteros et al., Sci Rep (2018)
- Chown et al., BES (2007); 21(2): 282-90