Your dependent variable is categorical so yes Logistic Regression is most likely appropriate here. Your independent variables are all categorical but you can still apply a logistic regression.
Your model looks something like this
$log(\frac{p}{1-p})= \beta_0 + \beta_1X_1 + \beta_2X_2$
where
$p$ is the probability of success in HIS101
$\frac{p}{1-p}$ is the odds of success in HIS101
$log(\frac{p}{1-p})$ are the log-odds of success in HIS101
$X_1$ is your IV1: Placement in Developmental English.
If the child is placed in Developmental English, then $X_1$ = 1, if not then $X_1$ = 0
$X_2$ is you IV2: Placement in Developmental Reading. If the child is placed in Developmental Reading, then $X_2$ = 1, if not then $X_2$ = 0
$\beta_1$ will tell you "How much do the log-odds of success in HIS101 change if the student is placed in developmental vs non-developmental Reading?"
$\beta_2$ will tell you "How much do the log-odds of success in HIS101 change if the student is placed in developmental vs. non-developmental English?"
If there are other variables that could confound your results, like demographic data on the child (i.e age, gender, race) or economic data (e.g family income, zip code), then it would probably help your analysis. A regression parameter $\beta_i$ tells you how much the log-odds of success changes for every unit change of $X_i$ holding all the other variables constant. So if a student is placed in Developmental English, $X_1 = 1$ and the log-odds of HIS101 success changes by $\beta_1$ for any given combination of Developmental Reading placement, age, race group, income level, and whatever else you put in the model. The assumption in the regression model is that all of these variables independently affect the probability of success. If you're looking at the impact of DevEng and DevRead on HIS101 success, you will have a stronger statistical argument of causation if you include more relevant variables in your model to control for their effects.
See also:
Regression with only categorical variables
