How can I omit the regression intercept from my results table in stargazer

I haven't your dataset, but typing omit = c("Constant", "x2") should work.

As a reproducible example (stargazer 5.2)

stargazer::stargazer(
  lm(Fertility ~ . , 
     data = swiss), 
  type = "text", 
  omit = c("Constant", "Agriculture"))

Edit: Add in omit.labels

mdls <- list(
  m1 = lm(Days ~ -1 + Reaction, data = lme4::sleepstudy),
  m2 = lm(Days ~ Reaction, data = lme4::sleepstudy),
  m3 = lm(Days ~ Reaction + Subject, data = lme4::sleepstudy)
)

stargazer::stargazer(
  mdls, type = "text", column.labels = c("Omit none", "Omit int.", "Omit int/subj"),
  omit = c("Constant", "Subject"),
  omit.labels = c("Intercept", "Subj."),
  keep.stat = "n")
#> 
#> ==============================================
#>                     Dependent variable:       
#>              ---------------------------------
#>                            Days               
#>              Omit none Omit int. Omit int/subj
#>                 (1)       (2)         (3)     
#> ----------------------------------------------
#> Reaction     0.015***  0.027***    0.049***   
#>               (0.001)   (0.003)     (0.004)   
#>                                               
#> ----------------------------------------------
#> Intercept       No        No          No      
#> Subj.           No        No          No      
#> ----------------------------------------------
#> Observations    180       180         180     
#> ==============================================
#> Note:              *p<0.1; **p<0.05; ***p<0.01

^{Created on 2020-05-08 by the reprex package (v0.3.0)}

Note the table should read. This appears to be a bug (stargazer 5.2.2).

#> Intercept       No        Yes       Yes    
#> Subj.           No        No        Yes