This can be accomplished in different ways:
1. BiGG has an API that you can access using various methods.
2. BiGG provides *.txt files that you can download.
Using either of these two methods will require you to write scripts to extract the information out. For e.g., if you have the enzyme names and want the corresponding BiGG identifiers of the reaction, you can parse through bigg_models_reactions.txt. However, often enzymes names might be written differently and you will not have an exact match. To overcome this, I will recommend using enzyme commission (EC) numbers, homologous gene names or gene symbols of the corresponding reactions to get the correct BiGG identifiers associated to the reactions.
Best,
Chintan